I prefer to wear as light an outfit as I can get away with for running, so vest, shorts and jockstrap are preferred - just enough for modesty and to keep my tackle secure. For very long runs I'm forced to swap the jock for compression gear, as my jock starts to chafe once it gets properly sweat soaked.
Setting aside any moral or legal considerations arising from jogging around town with an obvious boner, I'm surprised anyone can get away with running in shorts with no underwear full stop. Chafed nipples are agonizing enough, getting a chafed bell end doesn't bear thinking about.
The ultimate solution to that would, of course, be to dispense with the shorts as well and run bollock naked, as men did for countless aeons before clothes were invented - but opportunities to do that without ending up in a cell, padded or otherwise, are obviously somewhat limited...