Well, I am not playing the game for too long now, but I don't really see myself in these categories.
I would call myself a gamer, a gamer being a person who plays the game for the fun (and !!FUN!!).
I like these openended games, heroic last stands against the HFS, dragons, elephants and the like.
I like the adventure mode for simply being random, killing dragons but getting killed by horses.
While writing this, the word "casual" came up in my mind, but then it isn't the right description, since I play DF
in almost all my PC-time(some hours a day).
I do not take on big challenges like giant towers, nor do I want to complicate this game more by economy etc., I just like these "when you understood it, it is simple" kind of games.
Not implying that DF is simple, but you all remember this first "WTF" moment you had when looking onto DF the first time.
So, I would call myself a fungamer.
Not casual, but a person enjoying just playing for the sake of playing and getting weird events.
I play it to imagine a dwarf, standing on a retractable bridge over a volcano, shouting "YOU SHALL NOT PASS" at the onstorming army and sacrificing himself to hold them on the deathtrap long enough.
And then shouting "fly, fools" at the idiots who want to get his socks.
greetz, Freddy