I'm a Score Chaser; I'll play most any game, but he games I get really caught up in are the ones that rate the player based on speed and efficiency. Arcade-inspired games, basically, but also games like Metal Gear Solid and Super Metroid that aren't really about speed but lend themselves well to it. RPGs with myriad customization options/rules for combat that make combat either really challenging or a breeze also apply here, like Vagrant Story and SMT.

That being said, when I want to slow things down, I also really love games with diverse environments to explore. Tallon IV in Metroid Prime, various incarnations of Hyrule and Dracula's Castle, and FFXII's Ivalice are some of my favorites. Hell, I don't think it's a particularly good game, but I still remember Star Fox Adventures fondly for its environments and music.

Side Note: Kanno, I have a best friend who views games the same way you do, and I'd be lying if I said it hasn't rubbed off on me a bit. I think that's why I like games with scoring systems; it's feedback from the developers telling me that I'm playing the game the way they want me to.