I’ve been learning speedruns for the DS games in the Professor Layton series (so far I’ve completed runs in Diabolical Box and Unwound Future). However, I play on a homebrewed 2DS (no cheats or anything like that, just having the games digitally). I was wondering if that would still be listed under “emulator” or if there’s another subcat for it. I know some DS games have a “flash cartridge” category but I don’t know if that applies here.

You'll have to discuss this with the moderators of those specific games. I would not expect that to be considered emulation though, since you're still playing them on a real 2DS.

My only issue with that is that the game is extremely inactive so I’m not fully sure who to reach out to, or if anyone is even available.

The moderator for both the games you mentioned, Geshode, is active on here. There also appears to be a Discord server for the series.