i'm wondering if the OP can check the underside for a small, centrally located nob....that's the part where the baby plate would have been attached to a rock or skeleton of its mother. If that nob does exist, it's a fungia sp., if it doesn't it's a cycloseris sp.(i think that's how it goes! lol)
the upturned-at-the-edge growth may be just the actual expanded coral tissue. Cycloseris corals inflate quite a bit to help them move to find a better location (i just recently learned about this by virtue of seeing mine do the same thing). That is definitely the case with my cycloseris...and yes, the when mine inflates, it does 100% have the upturned-at-the-edge appearance.
also...cyclos and fungiids, shd be placed on the sandbed or on crushed coral on the substrate....they may do well glued to a higher rock, but i'd surmise they won't be happy in the long run. They are indeed motile corals...not sessile (cyclos at least).
z