Both Bachmann and Gingrich want tea party support. Gingrich is clearly an insider, taking over a million dollars to be a "historian" for Freddie Mack. He is clearly no Tea Partier. But he is unprincipled enough to claim allegiance to them if it would get him the nomination. He also might sell his granny into slavery for the same goal.
Bachmann is very close to insane and some sort of religious kook, as are some of the Teasies. She may be insane in a different way, though. There is no really clear definition of the Tea party, other than they despise Obama, think they are overtaxed, and demand a balanced budget. They oppose gay people in the military, oppose gay marriage, and most of them are strictly against abortions. They seem to have no clear plan for doing anything, though.
I am not sure that it is accurate to call Bachmann a Teabagger. She might call herself one.