A CATIIIb designation may have no decision height, or may have a decision-height that is 50ft or below. A CATIIIc designation has no decision height and no visbility requirements.
It is the individual carrier's OpSpecs and on-board equipment that determines approach capability, (a, b or c), not the regulator or ATC. Essentially there is no difference in ground equipment or runway holding position between CATIII a, b or c approaches.