I understand the persona role to be the fiction.
As I read Gygax's AD&D rulebooks, the key elements of the fiction are "You're in a narrow dungeon corridor", "You're in a dark room with a stream flowing through it", "You see a corroded tube in the stream", etc. Not "You're wearing yellow and are afraid of spiders".
"Moves" in the game then become things like "I walk down the corridor", "I light a torch to better get a look at the stream", "I poke the corrodied tube with my staff", etc. If you look at the example of play in Gygax's DMG, the closet we get to an expression of personality is the (unnamed) cleric describinb a giant spider as "nasty". I think that's a pretty generic personality.
But we get a lot of interaction with the shared fiction - poking things, climbing things, looking at things, trying to grab the spider and hurl it to the ground, etc - it's just that all that interaction is defined in functional terms: activities to which the players turn their characters.
I understand the functional role, so far as it is separable from the persona, to be the "gamey" element.
But in D&D the game doesn't
contrast with the fiction. Playing the game is
engaging the fiction. If you can define your game moves in purely mechanical terms, without reference to the fiction at all,
then it's not RPGing. D&D combat sometimes comes close to this (attack, damage, AC, etc are all defined in purely mechanical terms) but its terrain and movement/positioning rules involve adjudicating the fiction.
The core question of the poll and original post pretty much demands that we pick one. Basically, the only meaningful way I can read the question is, "What is role playing: a) focus on the fiction or b) focus on the game mechanics?"
The poll doesn't ask "What is role playing?" It asks "How do you primarily think of roleplaying?"
I'm not arguing that anyone is wrong to think of roleplaying in terms of personality. I'm explaining how it might also be thought of in terms of function/capability, and how that would nevertheless be RPGing as opposed to boardgaming, because it still means engaging with the fiction via a defined persona. It's not about who is
right, because it can be true for me that I think of RPing primarily as X, and for you that you think of RPing primarily as Y, and provided that X and Y aren't things that bear no connection to RPing, we might both make perfect sense.
The OP even went into some detail to elaborate the two ways of thinking about roleplaying, and to posit some ways in which the functional approach might bleed into the personality approach.
Building on those conjectures about "bleeding" in the OP, I'm also - but this is somewhat secondary - arguing that "function" is how "role" was presented in some early game texts, but that there is a change in that presentation somewhere in the mid-80s. That historical argument is interesting to me, but secondary to the premise and question of the thread.