I agree with the differences because of two reasons. This is pretty much the only time that I will get this techincal, LOL.
An invertebrates' dictionary or "textbook" defininition is: 'An animal lacking a backbone or notochord.......'. A notocord is 'a rod like cord of cells that forms the chief axial supporting structure of the body of the lower chordates, as amphioxus and the cyclostomes, and the embryos of the vertebrates'. Which invertebrates DO NOT have. Note hard corals have them. A mushroom is considered an Invertebrate because it lacks the ability to form a calcareous skeleton. Sometime this gets rather confusing because what about soft corals then. Well soft corals still have the ability to grow, reproduce and use calcium, in ways that invertebrates do not and never will.
I guess you could say that Corals have the ability to release calcareous deposits which in turn build a coral reef. You just don't really see a whole bunch of dead invertebrates leaving a reef behind.
As far as I know, there are only 4 classifications of creatures living in the sea. They definetely wouldn't be under Arthros' because they don't have jointed appendages or a hard exoskeleton. Couldn't be under Mollusks', because they do not have a soft unsegmented body (round is pretty segmented), a mantle, or a protective calcareous shell. They've already been tossed out of the vertebrate category because of numerous reasons, which leaves only one group.........I know i'm not taking the time to try to come up with a 5th, LMAO.