Well put Ber. By "professional" it means, as Ber put it, that the Dive Master, Dive Con, AI or Instructor (of whatever level) who pays insurance, dues & generally works with students, does refresher courses, Try or Discover Scuba sessions or acts as a dive leader (guide or such), what ever programs that are in the scope allowed for that professional level by their agency. Although it may open a can of worms, some professionals are paid, some are not, depends on the situation &/or the market they are working in.
Master Diver is a non professional level in which a diver must finish a certain curriculum (usually something to the effect of OW, AOW, Rescue, certain # of specialties, Certain # of logged dives). It is usually about the highest non- professional that can be reached within most agencies.