ok....a supplemental carrier is one who flys charters...a for-hire-type of deal. EAL, Pace, Airtours, etc....all charter type companies.
A flag carrier is anyone who flys under a U.S. flagged vessel. Most of the airlines whose pay rates you can search here are Flag carriers, albiet for the USA. A U.S. flagged carrier must be certified through U.S. code....49 U.S.C. 41102. That is also know as Title 49, subpart VII, part A, subpart ii, chapter 411 *41102. that section just says that the Secretary of Trans. MAY issue a certificate to any U.S. citizen who has applied to be a carrier. there are other things, but that is the jist of it.
Commercial is anyone "holding out". As an MEII, you know that. So you can tell yourself who in the airlines "holds out".
Domestic flys just that, domestic. No international. Southwest, Airtran, Jetblue, etc.
Sorry if i missed something. Have not dealt with this subject in a few years.
Hope this helped.