Could bullet chess and correspondence be considered as chess variants ? It seems to me that there is a strong case for this.

well the technical rules are exactly the same (that is, the board and pieces function in exactly the same way), I don't know if they should.

Granted, in bullet the player with the better mouse wins, and in correspondence you can use external databases etc and also move the pieces around, and you have a lot longer to think, but they're still the same game aren't they?

Just chess with crazy fast/really really slow time limits.

Well, yes, the rules for how the pieces move are the same, which can be said about 960 as well. But the rules of chess include other rules except the rules for moving the pieces. The use of external databases and books would violate the rules in "normal" OTB chess, as would moving pieces around on the board (or using another board), and such long thinking times would not be acceptable. Also, bullet chess, as played on the Internet, makes extensive use of pre-moves which are not permissible in OTB chess.