I agree that my version is wrong, but yours doesn't sound right either. ZFC doesn't have the standard integers as a model, or does it? I thought it also included other objects...

My version is right, but perhaps too restricted. The reason your argument works for ZFC is because it interprets PA by proving its axioms as applied to particular sets in ZFC. So the general requirement would be for a system to be strong enough to prove certain true statements about the natural numbers and to disprove certain false statements.

Could you give a reference?

No, I wrote nonsense - I realized that and wanted to come back and edit it pointing out this exact link you gave, but you did that before me. I don't know enough about Post's problem or the Friedberg/Muchnik solutions to say whether they can be suitably presented as provability classes.