the polyp is essentially the living portion of the coral. soft corals have tissues that extend beyond that to anchor themselves to substrates(rocks, sand, etc.) and stony corals deposit calcium carbonate skeletons. some corals like your yellow button polyps don't have much tissue to them aside from the polyp itself, like IslandCrow says, and are commonly referred to as just polyps. it's a name thing.
think of a polyp as sorta like a leaf on a tree. it's extending, trapping light, photosynthesizing, contributing to the life of the tree. if there was a leaf growing out of the ground you'd probably just call it a leaf. that's kinda like calling button polyps or zoanthids polyps. since those corals don't grow the same way as others, and don't come with 'trunks' or 'branches' they're sorta like a leaf growing out of the ground.