Uhhh yes you did state that they are anemones.
Mushrooms are sometimes referred to as "mushroom anemones" which is technically incorrect, they are not a type of anemone. Ndepratt is right -- they are coralimorphs.
Both anemones and corals are types of cnidarians, and both corals and anemones belong to the same subclass -- anthozoa. So they are closely related, but not the same thing. Jellyfish are the other major subclass of cnidarians.
A polyp is a term usually used to refer to one individual of a coral colony. So, a branch of SPS coral is made up of many hundreds of individual polyps. That picture of zoanthids you posted -- the whole thing is a colony, but each individual is called a polyp.
Hope that clears things up, and people should not be argumentative and defensive if they are proven wrong by subsequent posts! Just say "oops" and move on.