I would say any real RPG lets you define your character.
I think this is important, but I want to specify that I don't think it's important that this definition be
mechnical. It just needs to
be there.
Look at one of the simplest roleplaying games, not a video game but a verbal one - The Balcony. The game is simple - you are given a character, a name, a setting, a beginning, AND an ending. None of these things can be changed. All you can do is change the things you say, your character says - you fill the role, but you make it your own. You don't decide where you end up - but you decide who you are when you get there.
Most traditional RPGs are the same way - the beginning is set. Often times your character is set, in JRPGs especially, but sometimes it's simply your role and goal that is set. The ending is also, usually, set in stone - you have a lot of leeway on how to get there, but ultimately you end up in the same place.
What matter is the choices you make along the way, the way you
decide to react tot he world, the individual decisions you've made about who your character is, how they develop.
Personally, I'd consider games like The Walking Dead an RPG, in the truest meaning of the word. Your role is fixed - the beginning, the end, all fixed. What you do, the way you play the game, is
interpret that role - choose who your character is, what sort of person he's like.
Games like Final Fantasy are the same - the ending is always set in stone. All the major events are. The game is in your individual choices defining your character - what equipment they use, what they focus on, what they learn.
This is something that a shooter like Halo, for example, doesn't happen - there's no history, no progression. You, the player, don't make choices about how your character develops... and when you DO, like those shooters where you can choose to upgrade equipment and stuff, these are often considered to be "shooters with rpg elements".