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.

Authority is almost always used to describe argumentation based on sources other than logic or evidence.

The right kind of authority is based on logic and evidence. Most of most people' beliefs have not been personally verified by them. You probably havent' personally proven probability theory from scratch

Authority is practically meaningless as a concept if it includes both accurate and inaccurate foundations of argumentation

Fortunately, English allows us to qualify nouns with adjectives. Which allows us to distinguish between X-ish and Y-ish authorities.

. Probability theory works, appeals to authority don't.

Probability allows you to do things with clearly defined idea and evidence. Getting them clearly defined is the underwater part of the iceberg. That probability theory doens't help with.

Authority is almost always used to describe argumentation based on sources other than logic or evidence.

I don't think that's true. Most uses of "authority" are not about argumentation at all. The local parking authority, for existence.

Authority is practically meaningless as a concept if it includes both accurate and inaccurate foundations of argumentation. Probability theory works, appeals to authority don't.

I'm not sure this even makes sense enough to be wrong. I can't parse "accurate and inaccurate" with respect to "foundations of argumentation". Are you meaning to refer to fallacies, or something?

In general, there's nothing wrong with appeals to authority. It's well-understood that there is no formal logical step that takes one from "Authority says x" to "x". Nonetheless, TheOtherDave has it right:

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

It's worth remembering that other evidence screens off authority, but you have to take the evidence that you can get.