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[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.
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.
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.
Both are legal and acceptable.
Appendix C of the FIDE Laws of Chess describes the rules for writing the moves.
The relevant article if the c3 knight had not been pinned is C10 -
C.10 If two identical pieces can move to the same square, the piece that is moved is indicated as follows:
C.10.1 If both pieces are on the same rank by:
C.10.1.1 the abbreviation of the name of the piece,
C.10.1.2 the file of departure, and
C.10.1.3 the square of arrival.
but note that longer forms are acceptable. As article C8 says -
A longer form containing the square of departure is acceptable. Examples: Bb2e5, Ng1f3, Ra1d1, e7e5, d2d4, a6a5.
Only Ne2 is correct. There is a FIDE rule that explains what to do "if two identical pieces can move to the same square" (C10), that would lead to the Nge2 notation. But that rule simply doesn't apply here as the c3 knight cannot move to e2.
That said, there is no problem with writing Nge2, as it is still clear what move is meant. I think most software that reads PGN will also accept Nge2 (no guarantees!).
The correct notation stays Nge2 irrespective of the fact that Nce2 would be an illegal move in this position.
In such cases, always remember that the notation should be as complete and self-contained as possible, such that everything that happens in the game remains reproducible, even errors such as illegal moves. These things become more relevant when conflicts arise and an arbiter has to resolve it based on the record of the game. So by writing Nge2 you'll have left no room for uncertainty.
Chess notation serves merely as a way of recording the state and evolution of a game in a fully reproducible manner, it is not an implementation of the rules of the game per se.
Ne2 also remains technically correct in this case, I'm merely pointing out the importance of being on the safe side in such circumstances, leaving no room for doubt or false claims by opponent.