It depends on the slope of the roofing system and/or decking.
Type III will not run as readily as Type II, which is better to prevent asphalt drippage down the road.
BUT, a more viscuous product has more potential for self-healing any small imperfection type pin-hole leakage gaps.
So, the right answer, between the 2 types is what is it's intended end usage?
Coal Tar Pitch would be better than any of the more viscuous asphalts, btw.
Ed