Because of how sensitive an issue it can be, a brief explanation of the categories that appear in this site is in order.

Airline pilots are not the only ones who get caught up in titles, labels and definitions, however vane and petty they may be. As the saying goes, "a man will fight for pay, and kill for title". Attempting to define an airline, especially among the Regional and Commuter carriers can become a subjective exercise. Couple that with the way our lexicon can evolve over time, and you can have a mess. What was considered a regional airline in 1972, such as Southern Airways, would now fall under the category of Major / National airline. They were flying DC-9-15’s, but regionally.
Then there are those who point out the DOT definition, such as their definition of; Major Airline; US designation for an air carrier with annual operating revenue of more than one billion dollars, also called major carrier.

So after much research, I’m going to have to fall back on the definitions on Wikipedia which captures the essence of the various categories.

       Regional air taxi airline (carriers) fly and operate a fleet of aircraft with up to 9 passenger seats, and may or may not operate with its own airline brand or identity. Most usually aircraft are routed via the point-to-point on-demand or limited schedule transit models of aircraft routing.

       Regional commuter airline (carriers) fly and operate with their own ICAO or IATA codes, their own call sign, with a fleet of aircraft with 9-19 passenger seats, no flight attendant, non-stand up cabins, and function with or without an individual identity and brand, even though an individually operating certificated regional airline company. In many instances, the regional commuter airline may be owed by a conglomerate airline holding company, or may be a completely independent and individually owned regional commuter airline company. Most usually aircraft are routed via the point-to-point transit models of aircraft routing.

       Regional feeder-airline (carriers) fly and operate with their own ICAO or IATA codes, their own call sign, and fly and operate aircraft with 9-99 passenger seats, without an individual corporate identity or brand, even though an individually operating certificated regional feeder airline company. In many instances, the regional feeder airline may be owed by a conglomerate airline holding company, or may be a completely independent and individually owned regional feeder airline company. Most usually aircraft are routed via the hub-and-spoke model of aircraft routing.

       Regional airline carriers are federally operating certificated airlines which operate their own fleet of aircraft with their own brand, ICAO or IATA code, and their own call sign, upon a fleet of aircraft with 9-99 seats and always include at least one fleet of aircraft which requires a flight attendant. Typically aircraft are routed upon both the hub-and-spoke and the point-to-point transit models of aircraft routing. This category of airline carrier, being independent of regional feeder-airline arrangements with major airline carriers, typically are not shaped or size / growth constrained due to scope agreements  in place with mainline carriers which the feeder-airlines supplement.
Airline carriers operate their own fleet of aircraft with their own brand and identity, ICAO or IATA code, and their own call sign. Federally operating certificated regional airlines [9] are considered airlines. California Pacific Airlines is one of the newest airlines in the US, which observers may mistakenly call a regional airline because of its tendency to fly smaller under 100 passenger short haul aircraft.

      Major airline carriers operate their own fleet of aircraft with their own brand and identity, ICAO or IATA code, their own call sign, and operate at least one fleet of aircraft with more than 99 passengers upon their federal operating certificate. Major airlines have revenues greater than $1 billion  U.S. dollars in revenue during a fiscal year. Regional airlines are never considered major airlines although in many instances, the regional airline may be owed by a conglomerate airline holding company, and have revenues greater than $1 billion U.S. dollars in revenue during a fiscal year.

      Legacy Airline, an airline who can directly trace their roots to pre Air Mail Act of 1930

      National Airline,  in it’s pure form, means an airline that is the flag carrier for a nation, such as Air France.  However, in North America, it has taken on the meaning of a small, major airline.

      "The Big Four"  American, Eastern, TWA, and United.