It was more a description than a definition. Say what you will about most planes and the blurred lines, but there was a sharp dichotomy in the war between bombers and fighters. They really stood apart. The IL2 just didn't stand with the other bombers, IMO, with how it was regarded.
Horse hockey. You are thinking only in terms of US-type strategic bombing doctrine, which is not the end-all of what defines a "bomber." For instance, nearly every German bomber was designed for close-support of the ground forces. During the Battle of Britain they were
pressed into a strategic bombing role, for which they were ill suited. The IL-2 was purpose built in the same vein as the German bombers, for close support of ground forces, targeting ground targets. This is a BOMBER's role, plain and simple.
By your defiition, if the IL-2 is really a fighter, then the Stuka and JU-88 were really fighters as well. To make that claim is utter nonsense.