When people say polyps, they usually mean zoanthids and palys (palythoa and protopalythoa). They're classified as members of Order Zoanthidea, in Subclass Zoantharia.
Stony corals (what we call SPS and LPS) are in Order Scleractinia, which is also part of Subclass Zoantharia.
The critters we call mushroom corals (or mushroom anenomes) are members of Order Corallimorpharia, in Subclass Zoantharia.
So zoos, mushrooms and hard corals are in the same Subclass, but different Orders. Soft corals and gorgonians are members of a different Subclass, called Octocorallia.
Of course we couldn't be consistent, and only call members of Subclass Zoanthidea "polyps". We also use the term "green star polyps", which refers to a member of Subclass Octocorallia.
Clear as mud? Bob Fenner over at WetWebMedia.com does a very good job of explaining all the Classes, SubClasses, Orders, etc.