X is an authority with respect to a proposition P to the extent that X's assertion of P is evidence for P.

When someone uses "authority" in your sense #1, I understand them to be referring to something that claims to be an authority but is not actually authoritative. I agree that it's a very common usage though, especially in communities that are imprecise about their use of language. In such communities I might say "expert" instead, or "reliable source," or various other phrases that more-or-less mean the same thing but might differ in their connotations.