I just typed that question into google, this is the google reply: "Corals are different from anemone because they have a skeleton of sorts. Anemones are squishy and basically filled with water. Corals create a hard skeleton of calcium carbonate."
As you mention, the anemone is more resembling the soft coral. So whats the difference between hard and soft coral, according to google, "The biggest difference, as the name implies, is that hard-coral polyps form hard, calcium carbonate skeletons. Soft corals on the other hand are held together by a jelly-like mesoglea and rigid, spiny structures called sclerites hold together soft coral polyps"
Therefor, based on above, it seems that anemone is not a coral, because corals build a structure to help hold their bodies together, and anemone is mostly just water.