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The latter. Playing RTS games and poker have shifted me towards playing against the player a bit more, though not entirely.
I definately prefer to play the player, but once in a while I'll strike out and play the position. What about you?
I used to be a lot more oriented around "playing the player". I learned more trapping lines out of the opening and I'd sacrifice a pawn or two to complicate things while completely knowing that with semi-accurate play from my counterpart, I would lose. Now I'm moving more towards just trying to play the best move I can find because I think it's the next step up the ladder. However, once I go up that next step (meaning that I can find slightly better moves than I can now with more depth) it will be easier to spot moves that are equally complicating but less compromising for myself, and undoubtably, I will go back to playing the player 
I play the position, unless my opponent has time troubles that would be complicated by a more complicated but inferior move. I only wish I could play as Tal did.
Oops! I think I got mixed up with the name and what it meant. I do the first one, whatever that one is. Position, I think.
I play the player. Always. Even online, that's what I do and keep doing all the time.
I tend to play speculative sacs that mean nothing and pose no real threats sometimes- but which I know will throw my opponent off-balance. Still sometimes my sacs are objectively correct, but I don't really care because I'm quite confident that that's what I should do, and actually that's all I can do in chess right now, speculative sacs! Rooks and Bishops and Knights making their final move in the game just to take a pawn!
And it works!!!
Here's a game that I'm very proud of, I hope you see what I mean (BTW, my sacs here are justified, I think!):
I'm just a beginner, by the way, so I don't really have a "style". It's just how I play chess :)
2 Options. Choose 1.
Would you rather play the postition or play the player?
By this I mean, do you prefer trying to find the "most accurate" move you are capable of calculating or are you willing to play a deliberately inferior move because you know it complicates the position that you believe will confuse your opponent.
Carlsen is famous for the former while someone like Tal would be infamous for the latter.