I think it's the intersection, so, "and".

P(A|B,C) is "Probability of A, given both B and C".

That's true, may be a slightly confusing way to interpret it, given that C = A∪B. In context I think it's better to read it as P(A|B), in the case that C occurs.

wouldn't that be P((A|B)|C) ?

or even more precedence dubious: P(A|B|C) ?

The "given" symbol should only appear once in any probability statement.

That's true from an objective, mathematical standpoint. But the paradox is really saying something about how humans perceive the statement, so different (mathematically equivalent) ways of framing the same thing can make a difference. It's a minor point.

Yeah it's tricky because in most programming languages, '|' has precedence over ',', but in math sometimes it's the opposite. That tripped me up too.

In this context, the comma is as good as &.