It's the DM's world, and the characters are inhabitants of said world, and thus are a part of it. If the world (i.e. the DM) sees a character as alignment XY then that's what it is for game purposes, regardless of what the character or its player might think or want.
Put another way, if you want to be perceived as being of a given alignment then bloody well play to it, even if only vaguely and only most of the time. And there's many ways to play to each alignment, and if what you end up with is near a border then it'll reflect that way e.g. for a character who shows as somewhere on the border between CG and CN (which is probably the long-term average among al characters) I'll quietly write either Cg or Cn as its alignment; if it's played closer to pure N then it might show as cg in small letters, or whatever.
Put yet another way, if the player's play of the character says one thing and the written alignment on the sheet says another, one of those things has to change or else we're immediately into bad-faith play territory. (if you disagree with this we've nothing further to discuss; I've no time for bad-faith play or defense of such)
A DM who says "You're playing it wrong" is taking the stance that the play of the character has to change to get these two things back in synch; a DM who (like me) says "What's written on the sheet has to change" is in fact saying the player's been playing it right but has simply written the wrong letters on the character sheet.