I'll come from a slightly different angle.
1. Story- Gotta go with 1 here. 2 had some epic moments, but 1 is consistently good throughout, and I like the simpleness (I swear if someone says simple and clean)
+1 KH1
2. Gameplay- Going to have to disagree with you here. 2's gameplay is what makes kh2, kh2. It is the main reason people still play this game, and is considered the most well rounded, best combat to date.
+1 KH2
3.Worlds(explorability and design) - This really could be 2 different categories, but I'll combine them. And KH1 dominates them both. The simple little puzzles, and extras in KH1s world add a little flavor that you just don't see in 2. There's also just a lot more of an open feel, and platforming.
+1 KH1
4. Presentation- Gotta go with the flashiness of KH2 here.
+1 KH2
5. Soundtrack- KH2 is also a winner here. Many epic, and chilling songs
+1 KH2
6. Boss Battles- This one is very debatable in my view. You have some really really good boss design in 2, but 1 has the more consistent, fun bosses throughout. Still I gotta go with 2 here, just because they are designed a bit better
+1 KH2
7. Characters- I'm going to go with 1 here, just because I really liked the character development in 1 a lot better, and feel the group of friends, and the overriding goal. It was kind of cliche, with the best friend going bad, and Kairi as the damsel, but it still put it's own twist on it, and added a lot of emotion for me. The disney characters also have interactions in 1 that 2 didn't. It seemed like 2 was just Sora, Kairi, Riku, and Roxas. Though Roxas almost saved this category because his development was well done.
+1 KH1
8. Extras- Another very debatable one. Kh2 has COR. This alone could stand an argument, then you also have data mushroom, the cup, Terra, and Sephiroth. I still prefer 2's extras, just because there is a bit of depth to them. You have trinities, you have puppies. Just little things that add some variety next to the epic boss battles.
+1 KH1
9. Final World- Both of these have very epic final worlds, but I'm going to go with 2, just because it is a big boss rush, and you really feel the epicness. Though 1 you also feel a sense of having to save the world and combined with the music, would also make it a viable choice.
+1 KH2
10. FF/Disney aspect - Considering it's a crossover, I think this is important. Kh1 did the disney aspect better, Kh2 did the FF aspect better, but overall Kh1 had the best mix.
+1 KH1
I think these games are very interesting to debate, and to me are the 2 best. I do prefer 1 over 2, just because of the overall magic. It was interesting to see how the results turned out, and I may have even skewed the results. I also think 1 and 2 are hard to debate because they both do different things very well, and a case for either game could be made.
As for BBS vs DDD, ewww.