The current rules for 100% state "all ¤story¤ moons". Does this mean all distinct moons, or only the power moons which advance the story? Does this imply there's no limit to how many moons you can achieve by buying them?

In 100% you need to get all 880 unique moons, then buy more moons to get your moon count to 999

Those 880 moons are called "mission moons", distinguishing them from shop moons. There is also the (currently empty) category "All missions", which comes closer to the good old 120 star imo, but lacks the additional one moon per shop. Maybe replace said category with "All distinct moons"?

missions include the shop

Yes all missions include one moon from each shop, that makes 880, the next moons are just extra moons bought from the shop