Arguably “Casablanca” would fit: throughout the movie Rick is established as the guy who doesn’t get involved. (“I stick my neck out for nobody.”) By the end he’s re-defined himself. His speech to Ilsa at the airport would probably be the one to quote, though Rick voices his new insights as universal rather than as personal truths.
A good scene from Blade Runner:
SPOILER FOLLOWS FOR A 32 YEAR OLD MOVIE
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Not to mention when Gaff does the same thing to Deckard at the very end of the film.
Buzz’s delusions in Toy Story:
There must be a scene from Seconds that would fit the bill, but I couldn’t point you to a specific youtube clip.
The movie Buffy, The Vampire Slayer had a few …
“You’re not like other girls”
“Yes, I am”
“I have something those others did not have!”
[sneers] “Your faith?”
“My fine fashion sense.”
But it was your standard coming of age comedy of the time.
Streets of Fire had a lot of them, but they weren’t very positive …
Every version of Pride and Prejudice should have a moment of Elizabeth realizing she had been at fault, too, but I don’t think kids would appreciate it - and the scene is often not well done.
Well, not by that point, no. She did earlier in the series, though.
Thanks everyone, very helpful - except for you, Cartoonacy, who I do, in fact, pity. 
It’s the basic theme of *The Breakfast Club *-- remember the essay?
“All right people, we’re going to try something a little different today. We are going to write an essay, of not less than a thousand words, describing to me who you think you are.”
There’s a virtual parody of this sort of thing during the climax of Tropic Thunder when Robert Downey Jr’s character, an actor who lives his role 24/7 (and is playing a black guy throughout) is suddenly forced to look in a mirror. I know Ben Stiller movies aren’t to everyone’s taste but RDJ is excellently ludicrous in that film.
First thing that popped into my mind was the end of “Gone With the Wind” where Scarlett realizes that she loves Rhett. I don’t know if it’s exactly a “who am I” moment, though it’s certainly a personal awakening.
Jason Bourne: I can tell you the license plate numbers of all six cars outside. I can tell you that our waitress is left-handed and the guy sitting up at the counter weighs two hundred fifteen pounds and knows how to handle himself. I know the best place to look for a gun is the cab or the gray truck outside, and at this altitude, I can run flat out for a half mile before my hands start shaking. Now why would I know that? How can I know that and not know who I am?
A scene I loved from a movie I hated!
Joe vs. the Volcano. Joe (Tom Hanks) is a guy who’s always shied away from doing anything in life, and has just been told he’s terminally ill. With nothing to lose, he takes a lot of money from an eccentirc millionaire to carry out a crazy mission.
He decides to spend some of his money on clothes, and asks his chauffeur for advice.
The chauffeur understands, as Joe doesn’t, that the reall question being asked is “Who am I?”
Another “Who I am” moment from The Lion King.
Skip ahead a bit, to where Rafiki the mandrill drops a pebble in the water while Simba rests on a log.
*From Dusk Till Dawn *- outlaw George Clooney helps preacher Harvey Kietel resolve his crisis of faith. The scene more or less speaks for itself:
“I’m a mean… mmm mmm… servant of God.”
(Warning: Tarantino dialog).
Ha! Good one - “Me? I know who I am! I’m the dude, playing the dude, disguised as another dude!”
I actually stumbled upon the only way to truly enjoy this movie - the secret is…
Don’t watch the first 15 minutes of the film! It’s such a terrible beginning that it ruins your ability to appreciate anything that comes after it. I saw it for the first time on tv after having missed the horrible opening garbage, coming in only when Hanks starts his shopping trip in New York. Saw it, enjoyed it a lot more than I expected, and started to wonder why it had such a bad rap. Then, much later, I watched the whole movie with my wife who hadn’t yet seen it - she hated it, and I immediately figured out why, after suffering through that intro myself.
I think movies like From Dusk Till Dawn and Angel Heart are excluded by the OP’s conditions.