Go diving has a different connotation than go SCUBA diving. To go diving has the implication that you're jumping off the edge of a pool, a diving board, or a platform. To distinguish this from other types, you need the adjectives: SCUBA diving, free diving, sky diving, skin diving, etc.I've heard divers use the simple "go diving". That's probably the simplest and most common way to mention a diving trip.
In my experience, this is not true. No one participating in board or platform diving has ever left the house and said, "I'm going to go diving" (to my knowledge). Just as no sport parachutist ever talks about "going diving" to refer to leaping from a plane.To go diving has the implication that you're jumping off the edge of a pool, a diving board, or a platform.