Does playing Saturn mode on the HD remake count as Saturn or as HD? I assume Saturn but it's not clear from the rules.

It counts as HD. "Saturn mode" is only a graphical change.

So then what are the practical differences between HD and the original that cause them to be different categories?

I know this was four years ago, but I just noticed this so I'll answer your question now lol
HD is a remake and a completely separate game from the original. Basically every aspect and mechanic of the game is different in both versions. Here are a few differences:
In the original, movement is analog, so any direction you tilt the stick, NiGHTS will go in that direction. In the remake, movement is digital, so if you tilt the stick anywhere near one of the four cardinal directions, it'll force NiGHTS to go straight up, down, left, or right.
In the original, the kids incrementally speed up every time you jump in succession. (this has no speed cap.) You travel the same exact speed whether you jump or walk in the remake (unless you're walking up or down a slope.)
There is a gravity mechanic in the original. While traveling downward, you gain speed (with no speed cap.) This does not exist in the remake.
NiGHTS' movement speed is faster in the remake, and you spin around enemies that you've latched onto WAY faster.
There are tons more differences but I'm not going to write a whole book in this forum lmao. I'm thinking of making youtube videos about all the mechanics and speedrun strats in the original game at some point in the future.