I think ring as in NurburRING, Twin RING Montegi, SachsenRING , HungaroRING, NorisRING, the term RING refers to a track. It's where cars duel. The same would apply to Boxing ring- where boxers duel but not in the case of the Ring Cycle (Der Ring des Nibelugen) although the whole thing is about battling for the ring.
Ehm... I actually thought that "ring" in those names were derived from the fact that they're closed-loop tracks, thus somewhat "ring-shaped". Just my ignorance conjecturing.