They are. Though I'm guessing there's more to it than just putting on a green vest, I would assume he was credentialed and was actually an employee of a Jets-affiliated show. If someone noticed his camera pointed at the Pats sideline for the entire game, I think we'd have something here. But if he was filming here, there, everywhere, then there's nothing to gain by giving that tape to Rex.
The parts of the filming rules that are silly are the ones that got the Patriots busted because the filming wasn't done from a structure with a roof. (Why does it matter?) The part that allows team-affiliated shows to film, I have no problem with.