These two questions have been asked and, on consideration, are really interlinked. Answering one leads to the answer for the other. So I’ve chosen to do it that way, together. There is no general classification for aquariums as asked (as far as I am aware) so this text is just personal opinion.
Firstly, I suppose it is true to say that, strictly speaking, all reefs in aquariums are artificial. The only genuine reef is the wild one wholly created by Mother Nature out in the sea or ocean. However, things to my mind have become a little more complicated.
Accepting that all aquarium reefs are ‘artificial’ in the strict sense, I prefer to go a little deeper.