Recent reef formation has little to do with being zooxanthellate but depends on environmental factors. Using morphology to indicate the presence of zooxanthellae, there exist recent deepwater analogues to the shallow water azooxanthellate Devonian Edgecliff Bioherm. Here we have a concatination of terms, distribution, history, and ecology that makes this thread all the more intriguing. It can be located in shallow, deep, or intermediate depths. Neither has anything to do with corals or zooxanthellae specifically.
Coral reefs are bioherms or reefs made by corals and associated organisms.
Corals can have high cover on a volcanic substrate but that doesn't make it a coral reef in my opinion unless there has been carbonate accretion over the volcanic substrate. Catch-up reefs would also fit since they are usually grow into shallow water. Anyway, that is how I analyze the terminology based on first principles and dictionaries. Borneman's message is already displayed above. I concur wholeheartedly with Alina and would add a couple of small details.