So am I just supposed to beat the levels as fast as possible or do I have to collect all the colored coins in each level?

It depends on which category you are running. Look at the Categories / Variables. Get what you like, run it, stream it, (time it), submit it where it fits.

"Any%" in speedrunning always means "AFAP (as fast as possible)". So wihtout collecting anything or at least collecting less than intended. Basically just do what you want (if the rules allow it - a good example here is the BLJ in SM64 and each category) to get to the finish AFAP.

Longer Categories are titled "All Dungeons" (Zelda) or "Dark Side" (SMOdyssey).

And to get everything, even if it doesn't help you in-game, is called "100%", which could be considered "All Color Coins" in here.