I agree that heat ring and smoke ring are used to explain the same thing and that heat ring is more common, but........
I can find no evidence the ring has anything to do with heat in a thermal sense. It's presence or removal affects nothing but it's mass (as insignificant as it is) and the heat appears to stay the same in either case.
Now, when cooking on my wood stove if I add fresh wood, remove an eye and put a skillet over the hole I immediately appreciate the smoke ring which seals around the eye hole and keeps smoke traveling in the stove to the flue. Not leaking up around the skillet.
Thus, none of my skillets have heat rings, but all have smoke rings.
A woman can be called a girl or a lady. The queen should not be called a girl.
Hilditch