Live rock in its infancy for aquarium use was chunks of calcium carbonate seletons of dead coral that had broken off the reef and was collected for sale. It contained all of the critters that decided to call it home in the ocean.
I think the definition if there really is one has probably evolved, over the time since we have started to use it, to be closer to what Renee said above. However if one is paying a premium for live rock I would personally want the flora, fauna associated with rock harvested from the ocean. If I were only creating rock for bacteria I would save the buck and buy Tufa and throw some ammonia source in the tank and let the bacteria populate.