Pretty much what the title says. I was going to add some rhythm games I'm playing through, but true completion of the game won't happen for quite a while. Should I log progress for these longer "arcade" styled games?

Don't see why not, generally I'd do it like this

Assuming the game has multiple difficulties it would be:

Core plus a few extras = all songs beat on normal difficulty
Core plus a lot of extras = all songs beat on hard
Completionist = full completion

This is a little harder with something like Osu! which is basically all custom and way more difficulties but you'll have to salt to taste

Since it is pretty much entirely custom, I would consider something like Osu! to fall under "N/A" on the Incomplete - Completion scale, but for games that have a base track list shipped with it, I think @Lahdgren has a good suggestion above. In the end, there's no solid rule and it's up to your personal interpretation. :)

I loge very type of game that i play!

No, because I don't play rhythm games except for Audiosurf.