The government definitions don't take into account whether you sell your own tickets, but that is a key discerning factor for those who work in the industry. According to the Fed, Mesa is a major airline
I think these are more realistic definitions:
Majors: There are several flavors, but they all sell their own tickets (or package service), and mostly fly larger aircraft...
-Legacy: Extensive international service and have been around a while, UAL, AA.
-National/LCC: Domestic only, SWA, JB
-Big cargo: UPS/FDX
Regionals: Contract, don't sell many or any tickets
-Jet Regionals: Fly RJ's (SKW, AWAC)
-Commuters: T-props only (Colgan, Big Sky)
ACMI are kind of their own category. Startups such as VA and Skybus are not worthy of recognition at this point (and hopefully never will be
).