Those good, estabilished characters emerge naturally through game play, over the course of several sessions. That's why I don't let my players bring me huge backstories. I much rather leave things fuzzy at the beginning and let the characters be born and take shape of their own.
That said, in my experience the "that's my character would do" crowd are the least immersive, most disruptive ones. They tend to look more like forced caricatures than actual people.
Because, the vast majority of the time, "that's what my character would do" is an
excuse for doing what the player wants to do.
The ones that would sincerely say something like that won't say it as a petulant excuse for bad behavior. I find that, in the rare cases someone ever
does sincerely say something like that, it's because it's a moment of epiphany or a sad realization. It's not an
excuse for doing something that will upset others; it's the
discovery that what you would very much like them to do is something they absolutely are not going to do, or vice versa.
But that discovery is almost never about the kinds of things that would actively upset the other players, especially for a relatively long-running game. Hence, the only people who proffer up that excuse are, by and large, those who are trying to get away with something and
hiding behind the claim that they had no choice, the character "made" them somehow (despite this being obviously bunk.)
A good established character has certain independent existence. They cannot do whatever and remain in-character.
But that "certain independent existence" can be pretty slim, especially because patterns only exist so long as they aren't broken. Just any old break is usually not okay, but it's almost always possible to find
something that could be the start of development in a new direction.
Especially if...you guessed it...someone decides that that would be an interesting thing to explore. The character--the fictional construct--is not and cannot be capable of making decisions. Always, in every instance of things, it is the player making those decisions. For those who value immersion and continuity (note the difference between this and
consistency), a new decision needs to be rooted in past events and the evaluation thereof....but it is precisely that "the evaluation thereof" that permits change.
Otherwise, characters would be rigidly fixed in place, incapable of ever showing any growth or alteration whatsoever. As soon as you allow characters to
change in response to not just events themselves, but how the
player chooses to evaluate those events, you have ensured that it is not possible for the same character to be played the same way by two different people.
Earlier, I noted the difference between "continuity" and "consistency." Continuity is where that "certain independent existence" lies: there must be a set of
reasonable links between the things we knew (or thought) were true before and the things we know (or think) are true now. Continuity permits character development. Consistency, on the other hand, demands no change; two things that are not identical cannot be consistent with one another in the areas where they aren't identical. That's...kind of what consistency
means.
Absolute, impenetrable consistency is the death of character growth. But anything less than
absolute consistency necessarily relies on continuity--and continuity allows for different people to interpret the same set of inputs differently.