user19717: The definition of the back rank mate is: 

A Back-Rank Mate is a checkmate delivered by a rook or queen along the
  back rank in which the mated king is unable to move up the board
  because the king is blocked by friendly pieces (usually pawns) on the
  second rank.

For example white to move in this position 
 [fen "6k1/5pp1/8/8/8/3B4/5PPP/4R1K1 w - - 0 1"] 

King can't escape second rank, is this back rank mate?
What about this:
[fen "6k1/5bpp/8/8/2Q5/8/5PPP/4R1K1 w - - 0 1"]

What about this interesting black to move situation (its mate), is there a name for this mate?
[fen "6kr/5ppp/7N/2B5/8/6R1/8/1K6 w - - 0 1"]


user14340: First ask yourself why the differentiation of a 'back-rank mate' is meaningful. 
Generalized patterns that we call 'tactics' have names because it helps us to learn from them and respond to them. For example, I personally find the distinction between 'pin' and 'skewer' meaningful because it helps to separate the concepts of attacking 'through' either a less valuable enemy piece or a more valuable enemy piece. Likewise, understanding that pins with enemy kings on the other side are 'hard', and pins with enemy knights on the other side are 'soft' helps to make you wary of 'desperado' attacks that might help break your pin.
Really a back-rank mate could be seen as just a partially smothered mate. But it's important to see the distinction because of how common it is, so you can be aware of preparing for it. It is an acknowledgement of the fact that at the start of the game, your king only has 6 possible moves (being on the 1st rank), of which 3 are taken up by pawns, leaving you with just 2 open squares if your other pieces have developed already. Even after you castle, your king can remain vulnerable because it has only 2 squares of freedom, and both of those are on the same rank - meaning 3 of the opponent's pieces can checkmate your king with no support!
So, is it helpful to consider your examples above 'back rank mates'? I would argue yes. To me, being wary of back rank mates simply means I need to make sure my king has an 'escape path' when necessary; it is not so critical for me to say "back-rank pawn only" mate vs "back-rank my pieces only" mate vs "back-rank-2-of-my-pawns-and-by-the-way-a-bishop-is-attacking-h7" mate. In all 3 of those 'back-rank sub-types', I have failed to provide my king with enough mobility to avoid a common mating pattern.
Edit to consider your new example of mate with a knight, on the back rank. I would say that is not a 'back rank mate', because it does not involve an attack along the back rank, by the queen or a rook. Without that key element, the mating pattern is simply very different.

19717: <p>The <a href="https://en.wikipedia.org/wiki/Back-rank_checkmate" rel="nofollow noreferrer">definition</a> of the back rank mate is: </p>

<blockquote>
  <p>A Back-Rank Mate is a checkmate delivered by a rook or queen along the
  back rank in which the mated king is unable to move up the board
  because the king is blocked by friendly pieces (usually pawns) on the
  second rank.</p>
</blockquote>

<p>For example white to move in this position </p>

<pre><code> [fen "6k1/5pp1/8/8/8/3B4/5PPP/4R1K1 w - - 0 1"] 
</code></pre>

<p>King can't escape second rank, is this back rank mate?</p>

<p>What about this:</p>

<pre><code>[fen "6k1/5bpp/8/8/2Q5/8/5PPP/4R1K1 w - - 0 1"]
</code></pre>

<p>What about this interesting black to move situation (its mate), is there a name for this mate?</p>

<pre><code>[fen "6kr/5ppp/7N/2B5/8/6R1/8/1K6 w - - 0 1"]
</code></pre>
 14340: <p><strong>First ask yourself <em>why</em> the differentiation of a 'back-rank mate' is meaningful.</strong> </p>

<p>Generalized patterns that we call 'tactics' have names because it helps us to learn from them and respond to them. For example, I personally find the distinction between 'pin' and 'skewer' meaningful because it helps to separate the concepts of attacking 'through' either a less valuable enemy piece or a more valuable enemy piece. Likewise, understanding that pins with enemy kings on the other side are 'hard', and pins with enemy knights on the other side are 'soft' helps to make you wary of 'desperado' attacks that might help break your pin.</p>

<p>Really a back-rank mate could be seen as just a partially smothered mate. But it's important to see the distinction because of how common it is, so you can be aware of preparing for it. It is an acknowledgement of the fact that at the start of the game, your king only has 6 possible moves (being on the 1st rank), of which 3 are taken up by pawns, leaving you with just 2 open squares if your other pieces have developed already. Even after you castle, your king can remain vulnerable because it has only 2 squares of freedom, and both of those are on the same rank - meaning 3 of the opponent's pieces can checkmate your king with no support!</p>

<p><strong>So, is it helpful to consider your examples above 'back rank mates'?</strong> I would argue <strong>yes</strong>. To me, being wary of back rank mates simply means I need to make sure my king has an 'escape path' when necessary; it is not so critical for me to say "back-rank pawn only" mate vs "back-rank my pieces only" mate vs "back-rank-2-of-my-pawns-and-by-the-way-a-bishop-is-attacking-h7" mate. In all 3 of those 'back-rank sub-types', I have failed to provide my king with enough mobility to avoid a common mating pattern.</p>

<p><strong>Edit to consider your new example of mate with a knight, on the back rank</strong>. I would say that is not a 'back rank mate', because it does not involve an attack along the back rank, by the queen or a rook. Without that key element, the mating pattern is simply very different.</p>
