"We don't explore characters, we explore dungeons"
This Old School mantra is wrong in an interesting way.
I have a bunch of characters in a bunch of games and they're all the same and they're all different and none of that is on purpose.
They are all impatient schemers and sneaks who, nevertheless, try to make sure everybody in the party gets out alive. In the first hour it'd be almost impossible to tell them apart. All of them.
But Baron Blixa Von Apfelsaft (Thief lvl 8) is laid back, wary, and confident, Gorgut the Weasel (Elf lvl 2) is high-strung and boastful and likes to boldly announce "I am GORGUT! The WEAAAASEL!", Floyd Nine (Petty Call of Cthulhu Thief) is a self-pitying drunk, Sir Xyre of the Barrens (Pendragon Knight) is quick-tempered, prone to lawyerly evasion, and, when his fellow knights go courting, a fine wingman.
None of that was by design. Each one got there because they just shook out that way.
It probably goes without saying ('cause this is my blog) that all these characters are from Old School games with no personality mechanics or mechanical rewards for role-playing. This is just how Old School characters work: You get your PC a name, you get a few characteristics, you think of a voice, a few good things happen or bad things, and they suddenly seem competent or incompetent or skulky or brave and, without you doing anything, they begin to have a personality. Yet, because you're playing them, those personalities all have something in common.
Gorgut is Gorgut because (of necessity) he uses Unseen Servant and spare torches to trick gnolls and henchmen into thinking he's a great wizard, largely by shouting*, Floyd is Floyd because he's a Call of Cthulhu character so (of necessity) keeps getting the crap beat out of him, Blixa is Blixa because he's been around a long time--he started as a cynical Archeresque drunk, fought some fights, wised up, lost a beloved pet, got revenge, chilled out a bit, got micro famous for being in a lot of D&D games, etc. Their personality is either formed or revealed by circumstance--and the revelation is slow.
I think games with extensive personality mechanics expect that your character starts one way, undergoes Character Development in play (like they teach you in creative writing courses), and emerges another way.
With most games I like, the character starts no way at all, undergoes experiences which reveal character and then are proved to have been a certain way all along. Then, maybe, if they survive, undergo some character development.
Both of these ways of doing things show up in good movies and good books and real life. The first one you hear about more because it's tied into classic 3 act dramatic structure (a structure that serial fiction doesn't naturally have).
When people talk about remarkable moments in games with personality mechanics I often here them saying "Oh wow, I remember when you took that character there, I didn't see that coming, that was amazing." Moving the character along is a sometimes-conscious act on the continuum between writing and improv.
In the second model, you're not so much writing a character as performing an experiment on it and one of the many byproducts is finding out who the character is.
And that character is usually, unless you make an effort to do otherwise, an extension of a weird alternate you that comes out when you play games. But different parts of you depending on tiny factors This Alternate You + Elf + Blonde turns out different than This Alternate You + Cleric + Fat + Lost An Eye.
I like the artlessness of it. I'm not choosing a character to play. I am, literally, exploring the character, as one might a dungeon--going into it to see what is there. Not pushing it along, just knowing that I can dip a toe in at any time and see who somebody is.
*Hellwheel the Moonslinker has had 2 adventures, and is so far a lot like Gorgut The Weasel, but when he goes "I am Hellwheel the MOooONslinker!" there's, in the pronunciation of name alone, a level of irony that would go right over Gorgut's head.