I'm surprised anyone went for the first one, at all.
Well, I did!
I think it's a false dichotomy: you can't have a character without both function and personality. You can emphasize one and neglect the other, but you'd be falling short, and you can't ever really entirely eliminate either.
If I play my PC mostly around functions or capabilities (eg using spells if an MU, seeking to engage in melee combat if a fighter, etc) then a personality is likely to emerge - but the emergence of the personality is a byproduct of playing my role, not constitutive of doing so.
I think this is how the game was originally played. Personally, I think an approach of this sort still has advantages. It tends to reduce the incidence of "mere colour" in play (something I'm not a big fan of - if it's worth mentioning at all, then it should be mattering to resolution!), and it also reduces dissonance between mechanics and fiction (eg the character whose personality doesn't seem to match his/her stats).
I selected "personality" as an answer to the poll because it is that factor of personality which is present, in varying degree, that makes a game a role-playing game rather than a cooperative board game (i.e. that's the difference between Dungeons & Dragons, and Descent: Journeys in the Dark).
That's not the only difference.
The other difference, and I think the most important difference in the early days of RPGing, is that in a RPG
the fiction matters to resolution. In a boardgame, for instance, no matter how much the picture shows a wooden door, I can't declare that I set fire to it if that is not a defined move for the game. But in D&D I can.
It is also the presence of personality that allowed in the "old days" for two characters of identical class role to be distinctly different characters, at least to the degree that Nick's fighter and Greg's fighter didn't feel like interchangeable parts or duplicates while playing the game.
They also often differed in weapon selection, perhaps armour worn, and in any given encounter may have been allocated different tasks (eg holding the line vs defending the rear).
How does one resolve the question in a less structured narrativist game? Say a game like Monsterhearts, where the objective of the game is the creation of a personality that can drive dramatic tension?
I don't know Monsterhearts except by reputation. But I've been running quite a bit of Burning Wheel and also some Marvel Heroic RP lately, and so I did think about them in the context of the pell/thread.
Permit me a slightly roundabout way into the issue (!): upthread I noted that the 5e Basic PDF seems to frame the playing of the game in "functional" rather than "personality" terms, but that - unlike Gygax's PHB - it doesn't seem to emphasise "skilled" play very much. Rather than doing well by invoking and performing one's function, the idea simply seems to be to have fun doing so.
In BW and MHRP, I think the mechanics are intended to make personality/colour part of the character's function. (Hence mechanics like Beliefs, Distinctions, etc.) So performing the character's function will inevitably bring the character's personality to the fore, and perhaps lead to it changing (eg in my MHRP game, Nightcrawler ended up forsaking his Catholicism under Wolverine's more cynical influence, taking Mental trauma in the process). The colour of the characters is not
mere colour; and it is not a factor primarily just in free roleplaying or in choosing what action to declare. It matters to resolution.
I think MHRP is "light" in this respect - like 5e rather than Gygaxian AD&D - the player is expected to notice and enjoy the colour of the character feeding into, as well as emerging out of, play, but there is no real pressure to
do anything about it.
BW is more hardcore, the "indie" equivalent of Gyagxian skilled play: not only is personality/colour a key element of function, but the player is expected to work it hard, and there is definitely such a thing as doing it better or worse. Hence why I think BW can be quite a challenging game for players (far more demanding than 4e D&D or MHRP).
I don't know how Monsterhearts fits into that picture, but I hope the picture makes some sort of sense.