Aha! I found some suppliers that refer to "live rock" as "living rock:"
Gulf Specimen Marine Laboratory:
Rock Borer's Assortment - subtidal “living” limestone rock
Carolina Biological Supply:
Living Rock Community Set - rock alongside animals such as anemones, starfish, and crabs
Gulf Specimen Marine Laboratory:
Rock Borer's Assortment - subtidal “living” limestone rock
Carolina Biological Supply:
Living Rock Community Set - rock alongside animals such as anemones, starfish, and crabs