There used to be a differentiation between “Sport” and “Professional”, but the tools available to the professional divers (such as various gas mixes and multiple tanks) means the line is now blurred. “Serious” and “Casual”, seems to be valid nomenclature.
As for the actual depths, 30’ of water weighs the same amount as all the atmosphere above you, so at 30’ you experience twice atmospheric pressure, 60’ you have three times, 90’ you have 4 times. etc. I am not sure it is an arbitrary cut-off of based on a convenient point or there is an actual reason for it (I suspect the former) but 30’ is treated as a “safe” depth, and decompression is not required if you do not dive beyond this point. A lot of sport diving charters enforce a “every diver will make a decompression stop” rule just to be sure, as the tables were developed (as far as I know) by trial and error using human subjects by the US navy and not extrapolated from some basic scientific basis.
Note also the tables use a sea level base for their calculations. If you were diving in high altitude lakes or more commonly, flying after your dive, then you need to factor that in. A base rule of thumb is 12 hours between when it is safe to be at sea level and when it is safe to be flying.
Good luck with finding a base formula behind it all, but I am not sure one exists.