Yeah, they use it in a permanent revolution sense in order to build camaraderie with their viewers. The viewers aren't customers. They are fell participants in fighting against the media equivalent of the "man", even though Fox has grown far larger than all the "mans" in the media alone.

Do they honestly believe they are not mainstream or are they just doing it to pretend to separate themselves to gain a sense of "trust"?

I honestly think it depends on the host and individual employee. I'm pretty confident Ailes is a believer, and this would nudge the company culture that direction. Hannity is a believer. I don't think Megan Kelly is, but is using it as part of the "trust" branding.

I think O'Reilly represents a third possibility you missed. He comes across as someone who believes the media field is too liberal, but would also acknowledge Fox is part of the main stream media. He, back when I watched him, would often brag about how large Fox now was compared to the "main stream media". His usage seemed to be more of using "main stream media" as short form for "all the main stream media except for Fox".