To me, old school is inseparable from a very gamist paradigm. The whole "challenge the player, not the character" concept (with which I'm not sure I agree) translated into traps, tricks and puzzles, which I very cordially dislike in my gaming. I take that back. I don't dislike it cordially at all, I flat out refuse to engage them. They literally turn me right off a game I'm playing. You never forget, in old school gaming, that you're playing a game. It's structured at all times to be a game. This leads old school gaming to resemble very little the very fantasy stories that brought me into gaming and made the concept interesting to me, so I wandered away from old school gaming back when it was still "current school" gaming, making me one of the few prodigal D&D sons I know who actually dislikes First Edition AD&D quite a bit, and its concomitant paradigm of playstyle.
Old school also means a great deal of arbitrariness. Stuff happens, most frequently, because it was the result of some die roll on some table. Old school especially means dungeoncrawling. I hate dungeoncrawling. A lot. There's no interesting narrative to be pulled out of a dungeoncrawl.
You'd think, then, that I'd be a strong supporter of "new school" gaming; narrative control mechanics, stuff like that. As it turns out, I'm not that either. I prefer narrative to be what happens in game, not something that is dictated by metagame concerns, or managed by actual mechanics. But, I can't be interested in gaming that doesn't incorporate interesting plots, interesting characters and interesting scenarios. Plots does not imply pre-crafted railroad tracks on which the players must run their course without deviation. Plot is what happens after you've played. But that doesn't mean that reviewing your play sessions shouldn't more resemble the outline of a cool story than it does the diary of somebody just wandering around doing random stuff.
I don't know what that makes me. "Middle school?" Perhaps, there is no school that applies directly to my gaming tastes. Be that as it may, I do really enjoy the discussion about what makes gaming fun to different people. For me, I think I've always approached the hobby as a would-be writer. I want my characters to feel like well-drawn literary characters. I want my games to develop plots organically that would be interesting to read if they were converted to novel format. And yet, I want it to be a collaborative effort, with the players themselves driving the plot, not the GM. I came to gaming already a fan of fantasy literature, and good fantasy literature has always been my benchmark for gaming as well. I don't care too much about tactical depth of combat in the game. I don't care too much about puzzle solving (in fact, I strongly dislike most of it) and I do care about motivations, intrigue, and social difficulty, as well as, of course, plenty of rip-roaring action.