user11144: I bet I would play already 200 ELO better if I'd only check if my move isn't a complete tactical blunder. This obviously begs the question if engine cheating can be as easily detected if

I'd only use an engine that way
I'd constantly play only the second- or third-best move (of course there are forced moves, but it is probably unsuspicious to recapture the queen just traded...)
I only test critical decisions

After all, detection methods are meanwhile far more advanced than "plays the engine move". (I once managed to play the tablebase move for 10 or so moves in an endgame, but then, I'm very good at endgames...)
By the way, the questions would also be interesting for normal computer chess.
Disclaimer: You can tell me, as I'll never played online chess in my life and never will, especially due to the cheating possibilities. :-)

user27178: I'm afraid SmallChess's answer and comments above are more a matter of wishful thinking than actual fact. If you manage to catch a few small-time cheaters every now and then, it becomes easy to convince yourself that you can detect most or all cheaters; by definition, successful cheaters don't get caught, so you can pretend they don't exist. But they do, and perhaps more than you might expect. I know because for a long time i made my living developing and customising bespoke chess engines for this very purpose, and, properly used, over time, engine and (cheating) player become very much indistinguishable.
If you think about it, SmallChess is in effect proposing they can produce a robust Turing Test (distinguishing human from machine) that works even in the very restricted context of chess games (as opposed to free-flowing conversation, say). How likely do you think that is?
Believe you me, engine-assisted cheating is almost as widespread as doping in pro cycling.

user15223: tl; dr: the stronger you are, the more likely it is that you can cheat in this way and not get caught.
Check out this position:
[FEN "r2q1rk1/5p1p/p2p1bp1/4pn2/2B5/7R/1PPQ1PPP/2B2R1K w - - 0 1"]

This position is from Kasparov-Anand, Las Palmas, 1996. It's White's turn to move, and Kasparov went into a deep think. You might want to think about the position yourself, although if you are not a strong grandmaster, you probably won't see anything special.

At this point Kasparov went into a deep think. Jan Timman started to speculate whether White couldn’t play the very forceful 20.g4. Kasparov’s second Juri Dokhoian immediately confirmed: “That’s what he’s looking at!” Yuri understands Kasparov’s thinking better than anyone else in the world.
Meanwhile White had played 20.Bd5. The game lasted six hours, Anand defended very tenaciously and at around 10 p.m., much to the disappointment of Kasparov, a draw was agreed.
When he left the stage Garry spotted me and walked straight over. “I couldn't win it, could I, Fred?” he asked, with a troubled look on his face. It was a bit shocking: the world champion and best player of all times consulting a chess amateur, asking for an evaluation of the game he has just spent six hours on!
Naturally Garry wasn't asking me, he was asking Fritz. He knew I would have been following the game with the computer. “Yes, you had a win, Garry. With 20.g4!” My answer vexed him deeply. “But I saw that! It didn't work. How does it work? Show me.” He and Anand listened in horror while Juri dictated the critical lines. All of this was captured on video and published in ChessBase Magazine 56 (Feb 1997).

Part five of the series continues:

Most top grandmasters understand all too well how computers can affect the outcome of a game. In contrast to an amateur playing 600 points above his true strength, for whom the computer must dictate practically every move of the game, a strong grandmasters requires only occasional assistance to improve his performance dramatically. There are usually a few critical positions in which a player must decide whether a promising plan can work, or whether it is tactically flawed.
...
If we return to the example of the Las Palmas game given in part four, Kasparov’s second knew what he was looking at and actually uncovered the solution before the move was played. All that Kasparov needed was one bit of information: “Yes”. He didn’t need to get the message “20.g4! wins” but simply “There is a win” or even, in this specific case “The move we know you are looking at works!” That would have been enough to decide the game.

So yes: if you are a strong grandmaster, then having access to a computer telling you there is something in the position will result in a huge elo boost. You don't even need to know what the move is, you just need to know there is something. Furthermore this kind of cheating will not be easy to detect, since for almost all moves of the game you will do just fine on your own.
On the other hand, if you are an amateur, then you will need help from the computer much more often, and furthermore you'll need specific help. Look again at the position: would knowing that White has a win have helped you at your level? As you mentioned, detecting cheating is by matching one's moves against the computer's. Naturally, once you have to consult the computer more often, cheating becomes easier to detect.
Edit: Quoting from this source on the kind of cheating you propose:

Faced with a complex calculation, a player could sneak their smartphone into the bathroom for one move and cheat for only a single critical position. Former World Champion Viswanathan Anand said that one bit per game, one yes-no answer about whether a sacrifice is sound, could be worth 150 rating points.
"I think this is a reliable estimate," says Regan. "An isolated move is almost uncatchable using my regular methods."

[Regan is an academic who's developing methods to catch cheaters.]
But Regan has ideas on how to catch this kind of cheating. They are untested, but he's confident they'll work. See the source for more.

user825: Yes. You will still be detected just that the site is going to need more games. You are more likely to stay under the radar, but that doesn't mean you won't be caught. Cheat detection has never been just comparing your moves against the computer.
To understand, online cheat detection is an application of anomaly detection in data science (google "Anomaly Detection" if you need help). Chess.com calculates dozen of metrics such as the probability of beating a higher rated opponent due to luck, and many other metrics that they will never tell you. They are also going to do it for everybody in the database. Once they do it, they will then have a distribution of the metrics for each rating group.
If you are a smart cheater, you will win games slower than a dumb cheater but still more statistically significant than everybody else not cheating. Chess.com will create a mathematical model such as hypothesis testing to derive a p-value probability of you cheating given a null hypothesis that you aren't a cheater. As you win more and more games, the standard errors of your profile will get smaller and smaller, your p-value will also get smaller. It will just be a matter of time until chess.com bans your account.
Even if you don't end up winning, your in-game performance such as your ability to stay consistent with the top-three computer moves or your ability to play time-pressure high complexity positions while not blundering will be statistically different from everybody else in your rating group. Chess.com will be able to detect you even if your moves don't match with the computers.
That's exactly how you see why cheaters like Dewa Kipas were able to cheat for a few weeks until his match with IM Levy Rozman. (https://www.chess.com/news/view/most-watched-chess-stream-in-history-dewa-kipas). Chess.com needed time and data to build up Dewa Kipas's cheating portfolio.
You will be caught given a high enough number of games, but a single game won't be enough.

11144: <p>I bet I would play already 200 ELO better if I'd only check if my move isn't a complete tactical blunder. This obviously begs the question if engine cheating can be as easily detected if</p>
<ul>
<li>I'd only use an engine that way</li>
<li>I'd constantly play only the second- or third-best move (of course there are forced moves, but it is probably unsuspicious to recapture the queen just traded...)</li>
<li>I only test critical decisions</li>
</ul>
<p>After all, detection methods are meanwhile far more advanced than &quot;plays the engine move&quot;. (I once managed to play the tablebase move for 10 or so moves in an endgame, but then, I'm very good at endgames...)<br />
By the way, the questions would also be interesting for normal computer chess.</p>
<p>Disclaimer: You can tell me, as I'll never played online chess in my life and never will, especially due to the cheating possibilities. :-)</p>
 27178: <p>I'm afraid SmallChess's answer and comments above are more a matter of wishful thinking than actual fact. If you manage to catch a few small-time cheaters every now and then, it becomes easy to convince yourself that you can detect most or all cheaters; by definition, successful cheaters don't get caught, so you can pretend they don't exist. But they do, and perhaps more than you might expect. I know because for a long time i made my living developing and customising bespoke chess engines for this very purpose, and, properly used, over time, engine and (cheating) player become very much indistinguishable.
If you think about it, SmallChess is in effect proposing they can produce a robust Turing Test (distinguishing human from machine) that works even in the very restricted context of chess games (as opposed to free-flowing conversation, say). How likely do you think that is?</p>
<p>Believe you me, engine-assisted cheating is almost as widespread as doping in pro cycling.</p>
 15223: <p><strong>tl; dr: the stronger you are, the more likely it is that you can cheat in this way and not get caught.</strong></p>
<p>Check out this position:</p>
<pre><code>[FEN &quot;r2q1rk1/5p1p/p2p1bp1/4pn2/2B5/7R/1PPQ1PPP/2B2R1K w - - 0 1&quot;]
</code></pre>
<p>This position is from Kasparov-Anand, Las Palmas, 1996. It's White's turn to move, and <a href="https://en.chessbase.com/post/a-history-of-cheating-in-chess-4" rel="noreferrer">Kasparov went into a deep think</a>. You might want to think about the position yourself, although if you are not a strong grandmaster, you probably won't see anything special.</p>
<blockquote>
<p>At this point Kasparov went into a deep think. Jan Timman started to speculate whether White couldn’t play the very forceful 20.g4. Kasparov’s second Juri Dokhoian immediately confirmed: “That’s what he’s looking at!” Yuri understands Kasparov’s thinking better than anyone else in the world.</p>
<p>Meanwhile White had played 20.Bd5. The game lasted six hours, Anand defended very tenaciously and at around 10 p.m., much to the disappointment of Kasparov, a draw was agreed.</p>
<p>When he left the stage Garry spotted me and walked straight over. “I couldn't win it, could I, Fred?” he asked, with a troubled look on his face. It was a bit shocking: the world champion and best player of all times consulting a chess amateur, asking for an evaluation of the game he has just spent six hours on!</p>
<p>Naturally Garry wasn't asking me, he was asking Fritz. He knew I would have been following the game with the computer. “Yes, you had a win, Garry. With 20.g4!” My answer vexed him deeply. “But I saw that! It didn't work. How does it work? Show me.” He and Anand listened in horror while Juri dictated the critical lines. All of this was captured on video and published in ChessBase Magazine 56 (Feb 1997).</p>
</blockquote>
<p><a href="https://en.chessbase.com/post/a-history-of-cheating-in-chess-5" rel="noreferrer">Part five of the series continues</a>:</p>
<blockquote>
<p>Most top grandmasters understand all too well how computers can affect the outcome of a game. In contrast to an amateur playing 600 points above his true strength, for whom the computer must dictate practically every move of the game, a strong grandmasters requires only occasional assistance to improve his performance dramatically. There are usually a few critical positions in which a player must decide whether a promising plan can work, or whether it is tactically flawed.</p>
<p>...</p>
<p>If we return to the example of the Las Palmas game given in part four, Kasparov’s second knew what he was looking at and actually uncovered the solution before the move was played. All that Kasparov needed was one bit of information: “Yes”. He didn’t need to get the message “20.g4! wins” but simply “There is a win” or even, in this specific case “The move we know you are looking at works!” That would have been enough to decide the game.</p>
</blockquote>
<p>So yes: if you are a strong grandmaster, then having access to a computer telling you there is <em>something</em> in the position will result in a huge elo boost. You don't even need to know what the move is, you just need to know there is something. Furthermore this kind of cheating will not be easy to detect, since for almost all moves of the game you will do just fine on your own.</p>
<p>On the other hand, if you are an amateur, then you will need help from the computer much more often, and furthermore you'll need specific help. Look again at the position: would knowing that White has a win have helped you at your level? As you mentioned, detecting cheating is by matching one's moves against the computer's. Naturally, once you have to consult the computer more often, cheating becomes easier to detect.</p>
<p><strong>Edit</strong>: Quoting from <a href="https://cse.buffalo.edu/%7Eregan/personal/JuneCLarticleKWR.pdf" rel="noreferrer">this source</a> on the kind of cheating you propose:</p>
<blockquote>
<p>Faced with a complex calculation, a player could sneak their smartphone into the bathroom for one move and cheat for only a single critical position. Former World Champion Viswanathan Anand said that one bit per game, one yes-no answer about whether a sacrifice is sound, could be worth 150 rating points.</p>
<p>&quot;I think this is a reliable estimate,&quot; says Regan. &quot;An isolated move is almost uncatchable using my regular methods.&quot;</p>
</blockquote>
<p>[Regan is an academic who's developing methods to catch cheaters.]</p>
<p>But Regan has ideas on how to catch this kind of cheating. They are untested, but he's confident they'll work. See the source for more.</p>
 825: <p><strong>Yes</strong>. You will still be detected just that the site is going to need more games. You are more likely to stay under the radar, but that doesn't mean you won't be caught. Cheat detection has never been just comparing your moves against the computer.</p>
<p>To understand, online cheat detection is an application of <strong>anomaly detection</strong> in data science (google &quot;Anomaly Detection&quot; if you need help). Chess.com calculates dozen of metrics such as the probability of beating a higher rated opponent due to luck, and many other metrics that they will never tell you. They are also going to do it for everybody in the database. Once they do it, they will then have a distribution of the metrics for each rating group.</p>
<p>If you are a smart cheater, you will win games slower than a dumb cheater but still more statistically significant than everybody else not cheating. Chess.com will create a mathematical model such as hypothesis testing to derive a p-value probability of you cheating given a null hypothesis that you aren't a cheater. As you win more and more games, the standard errors of your profile will get smaller and smaller, your p-value will also get smaller. It will just be a matter of time until chess.com bans your account.</p>
<p>Even if you don't end up winning, your in-game performance such as your ability to stay consistent with the top-three computer moves or your ability to play time-pressure high complexity positions while not blundering will be statistically different from everybody else in your rating group. Chess.com will be able to detect you even if your moves don't match with the computers.</p>
<p>That's exactly how you see why cheaters like Dewa Kipas were able to cheat for a few weeks until his match with IM Levy Rozman. (<a href="https://www.chess.com/news/view/most-watched-chess-stream-in-history-dewa-kipas" rel="noreferrer">https://www.chess.com/news/view/most-watched-chess-stream-in-history-dewa-kipas</a>). Chess.com needed time and data to build up Dewa Kipas's cheating portfolio.</p>
<p>You will be caught given a high enough number of games, but a single game won't be enough.</p>
