user16626: [FEN "rnbq1rk1/ppp2ppp/4pn2/3p4/1bPP4/2NBP3/PP3PPP/R1BQK1NR w KQ - 0 6"]

If white move his king knight to e2 in the next move, should this move be recorded as Nge2 or Ne2 in the algebraic notation? 
After all, the other knight cannot legally move to e2 in this move.

user10380: It should be Ne2 because, as you correctly say, only that knight may move to e2.
By contrast, if, say, each of two knights could move to e2, but one would discover check and the other would not give check, then the move must be disambiguated in both cases. The presence or absence of a + does not disambiguate. Cf, from the PGN reference:

8.2.3.5: Check and checkmate indication characters
If the move is a checking move, the plus sign "+" is appended as a suffix to
  the basic SAN move notation; if the move is a checkmating move, the octothorpe
  sign "#" is appended instead.
Neither the appearance nor the absence of either a check or checkmating
  indicator is used for disambiguation purposes.  This means that if two (or
  more) pieces of the same type can move to the same square the differences in
  checking status of the moves does not allieviate the need for the standard rank
  and file disabiguation described above.  (Note that a difference in checking
  status for the above may occur only in the case of a discovered check.)

Cf this question.

user12523: It doesn't matter. The purpose of algebraic notation is to produce an unambiguous record of the game, not a canonical representation that you could compare character by character.
This is not spelled out on the official rules, but I believe it is the inescapable conclusion considering that some parts of the notation are explicitly optional (here I'm talking about the FIDE Laws of Chess); for example:

[...] each player is free to use the name which is commonly used in his country [...]
A longer form containing the square of departure is acceptable.
[...] an x may be inserted [...]
[...] ‘e.p.’ may be appended.
[...] The last four [capture, check, checkmate, and en passant] are optional.

That said, the rules are clear that disambiguation is only needed when two identical pieces can move to the same square, so the "correct" notation would be Ne2.

16626: <pre><code>[FEN "rnbq1rk1/ppp2ppp/4pn2/3p4/1bPP4/2NBP3/PP3PPP/R1BQK1NR w KQ - 0 6"]
</code></pre>

<p>If white move his king knight to e2 in the next move, should this move be recorded as Nge2 or Ne2 in the algebraic notation? </p>

<p>After all, the other knight cannot legally move to e2 in this move.</p>
 10380: <p>It should be Ne2 because, as you correctly say, only that knight may move to e2.</p>

<p>By contrast, if, say, each of two knights could move to e2, but one would discover check and the other would not give check, then the move must be disambiguated in both cases. The presence or absence of a <code>+</code> does not disambiguate. Cf, from the PGN reference:</p>

<blockquote>
  <p>8.2.3.5: Check and checkmate indication characters</p>
  
  <p>If the move is a checking move, the plus sign "+" is appended as a suffix to
  the basic SAN move notation; if the move is a checkmating move, the octothorpe
  sign "#" is appended instead.</p>
  
  <p>Neither the appearance nor the absence of either a check or checkmating
  indicator is used for disambiguation purposes.  This means that if two (or
  more) pieces of the same type can move to the same square the differences in
  checking status of the moves does not allieviate the need for the standard rank
  and file disabiguation described above.  (Note that a difference in checking
  status for the above may occur only in the case of a discovered check.)</p>
</blockquote>

<p>Cf <a href="https://chess.stackexchange.com/questions/1864/is-this-case-considered-an-ambiguity-or-not">this question</a>.</p>
 12523: <p>It doesn't matter. The purpose of algebraic notation is to produce an unambiguous record of the game, not a canonical representation that you could compare character by character.</p>

<p>This is not spelled out on the official rules, but I believe it is the inescapable conclusion considering that some parts of the notation are explicitly optional (here I'm talking about the FIDE Laws of Chess); for example:</p>

<blockquote>
  <p>[...] each player is free to use the name which is commonly used in his country [...]</p>
  
  <p>A longer form containing the square of departure is acceptable.</p>
  
  <p>[...] an x may be inserted [...]</p>
  
  <p>[...] ‘e.p.’ may be appended.</p>
  
  <p>[...] The last four [capture, check, checkmate, and <em>en passant</em>] are optional.</p>
</blockquote>

<p>That said, the rules are clear that disambiguation is only needed when two identical pieces <em>can</em> move to the same square, so the "correct" notation would be Ne2.</p>
