I agree that diving MAY not be a Game, but it is definately a sport. It is a physical activity done for enjoyment. An while there is a relaxation element to diving, for many the relaxation is part of the challenge. That is puspose of the relaxation is not an end in its self, it is a means to extend their depths and bottom times. They are competing with themselves to reduce their SAC in much the same way a runner compete with themselves to decrease their time in a 10k. Many divers spend a lot of time cycling, swimming, an hitting the gym on a relular basis so that they can reduce the SAC.
So if you are measuring your SAC and training to reduce your SAC, is that not competition? How is this different from a running measuring their personal best in the 10k? In that sense is it a game? What if two divers compare their SAC after a dive?
However, if your are diving to relax as a source of relaxation as an end in itself, well that fits the defination of hobby a bit closer. And there are many people that dive as a source of relaxation. I would imagine that these are the majority of divers, but this is not all divers for sure.
So, I think that the answer to this question is much the same as a lot of questions:
It depends.