In my opinion the fundamental difference is whether you grew up with the music around you, or whether you discovered it later. If you're in the former category, you're a traditional singer no matter how many songs you subsequently learn from Martin Carthy albums; in the latter case you'll never be truly "traditional", no matter how many "old boys" you learn songs from.

I think much of the reluctance to accept this distinction comes from a romantic wish to be "part of the tradition".