I'm with JSH on this. The 'regional' is the subcontractor. The RFP issued by the 'mainline' carrier specifies everything from the amount the regional carrier will be paid to fly the routes to the paint schemes the airplane will wear. The 'regional' bids for the right to provide the service per the RFP's requirements. The 'regional' carrier serves at the pleasure and whims of the mainline partner. Go find an ex-Freedom Airlines manager if this seems confusing and ask them what the relationship was between F8 and Delta.

I feel like I'm missing something here in your argument, 727574......but having read a few of these agreements in my time in the industry (in a job function other than line pilot), I'd say that in the example of a mainline carrier and a regional carrier, the 'mainline' definitively is the customer and the 'regional' carrier the provider of the service.