Who Green Lantern III is is really debatable. Guy was technically introduced first, and used the power ring before John, but he was an infrequent supporting character and not REALLY the Green Lantern (at least not from a certain point of view). Hal Jordan was still Green Lantern, and Guy wasn’t really a full fledged Corps member.

Eventually, Guy was incapacitated and John filled Guy’s spot as the understudy. The big difference, though, is that in the ’80s, John became the full time and primary Green Lantern of Sector 2814, and actually was the star of the “Green Lantern” comic for a good length of time. So, John undeniably became “Green Lantern,” and not just an infrequent supporting understudy. Guy Gardner never really was Hal Jordan’s replacement. Hal’s first true replacement was John Stewart.

Guy didn’t become a full time Green Lantern until after John already was one. It’s one of the sources of his anger. He felt that he missed his chance at being Green Lantern twice. The first time was because Hal Jordan was in closer proximity to Abin Sur’s crash site, so the ring chose Hal Jordan, and the second time was because he was in a coma (which was actually somewhat Hal Jordan’s fault), and John got the ring instead. Also, while Guy was in a coma, Hal Jordan started a romantic relationship with Guy’s girlfriend, Kari Limbo.

My view is that John is Green Lantern III and Guy is Green Lantern IV, because, despite being introduced first, the mantle of Green Lantern never fully went to Guy until after it had already gone to John.