Don't quote me on it, but I am pretty sure that the large majority of elite German parachute troops that the Americans encountered in the War, were mainly attached to SS divisions. By 1944, the more elite elements of the German army had been depleted so badly that forming full divisions was not possible, so they reinforced the SS. You can watch all the documentaries/movies and see parachute troops fighting alongside SS infantry divisions, and occasionally with the Wehrmacht. Downfall is a good example of this, in the defense of Berlin.
In short, for the Americans, most of the fallschirmj?gers were probably just called fallschirmj?gers, or mistakenly called SS troops. Green devils was a term more commonly used by the Brits and French, if my sources are correct.
-Blue