User blog comment:TumblrAme/Natsu's Team/@comment-4449950-20141218012217/@comment-9595671-20141222065512

But Lucy considered the team broken up after the Everlue thing due to Everlue wanting a blonde maid.

I don't think the team was truly official until Erza mentioned they had worked together often enough after the Phantom Lord arc. So I think that's a better place to think around the team name, for all we know they may have actually called themselves Team Fairy Tail like their GMG name. (minus the A of course)