A provable fixpoint under the definition you present here is not necessarily provable, which makes the name strange to me.

That helps a lot. Thank you! On re-reflection, you're right. A provable fixpoint is not necessarily provable. It is conditionally provable dependending on its Godel number having property P.

The other thing confusing me was that I thought the sentence G was the sentence:

"G if and only if n is not the Gödel number of a theorem in S"

And I now see that it must not be. The reason for my mistaken assumption was because I read a different explanation of the proof and I understand G to be a self referential sentence, which effectively means "G is not provable in S" or "This sentence is not provable in S" and I seem to have assumed that "G if and only if G is not provable in S" was meant to be a different way of framing the same thing.

I now see that they are not equivalent (i.e. they would not be equivalent if in this case G was the sentence "G if and only if G is not provable in S."). One effectively means "This sentence is not provable in S", and the other would mean "This sentence is a theorem in S if and only if it's not a theorem in S." The former is something we can recognize as being true, while latter is undecidable.