Mason,
I think you're splitting rabbits. AMR is a holding company. It produces no product, nor does it sell anything to anyone (except maybe stock to investors).
American Airlines is the 'brand name' airline that sells tickets and cargo space.
American Eagle is a lift subcontractor that has no capacity to sell its product to the general public.
The distinctions are pretty clear to all of us who are familiar with the airline business.
It shouldn't be that hard to differentiate those airlines who sell to the public, and those who sell to other airlines. If the term 'Major' is no longer adequate how about 'Real' airlines and 'Subcontrator' airlines?