You're thinking about Fallout.
I don't think you're missing anything, as much as having a different definition of "role-playing game" from those who label Diablo as such. I think the idea is that you are playing a role in that every Diablo character is (theoretically) different depending on the choices of the player. While I don't necessarily agree with the definition, it's probably too established to change much (though it feels like descriptions like "action game with role-playing elements" or whatever is getting more common).