Plate Coral and Disk Coral are usually used to describe the same coral, usually referred to as Plate Coral.
Heliofungia only has one species, in our hobby, Heliofungia actiniformis, and there's confusion if "Heliofungia" is even an appropriate name, or if they're all Fungia corals. Heliofungia actiniformis was originally named Fungia actiniformis.
Either way, they all have the ability to "move," though it's not all that common. So, just because it doesn't move, doesn't mean it's not one or the other.