The term was originally used very early in the game to describe the edges of features. The fringe of the green, the fringe of the trees etc. It's not clear if the fringe of the green was cut at a different height.
When mowing was powered by pushing the mowers, sometimes these areas were cut with the same mower, but not as frequently, and the difference in frequency of cut left the grass taller. This is just a guess based on pictures, but I think some examples of features that were cut by hand like this were the mounds behind 12 at GCGC, and the first half of the bairritz green through the swale at Yale and possibly others.
It is unclear to me when the practice began of mowing a separate fringe, apron, or collar cut around the green at a higher level of cut. But were I to place it on a timeline, I would start around 1927-1930 when Toro and Jacobsen began making a good reliable lightweight power mower. And then I think it would have been put into practice for false fronts, turnarounds between bunkers and greens, approach transitions between the green and the tractor drawn gang mowers. Surely when maintenance became more mechanized some new traffic problems arose around the green that the fringe cut resolved.