In the case of arithmetic "it's true" means "it's true in the standard model". I don't know anyone else who uses it differently. Likewise, for any theory that has a standard model, whenever someone says it's true relative to that theory, they almost always mean true in the standard model of the...
I know this is super old school, but in case someone else finds this thread, I thought it might be worth clarifying this highly interesting discussion.
It's true that if GC is independent of PA then it's true. It's also true that PA + ~GC is consistent, but that's irrelevant to the previous...