"True" in this context is a technical definition, not some hand-waving thing. A sentence X being true means every model satisfies X. Provable means logical deduction within the theory proves X. So actually, Godel's theorems are exactly about truth and provability.
Yes if you appeal to model theory you're on firm ground with truth. But that's not how most informal introductions to the incompleteness theorems go (and indeed not how this article went about it).
More to the point, Godel's incompleteness theorems are theorems of proof theory, not model theory. Their proofs can be carried out with no recourse to models at all. And I think they are far less prone to being misunderstood when carried out without recourse to models and by extension without recourse to semantic truth.
You can of course choose to interpret them in model theory if you're working in a semantically complete logical system such as first-order logic with the usual semantics or second-order logic under Henkin semantics, and it is often illuminating to do so once you have those fundamentals. But I think in an informal setting those will tend only to confuse rather than clarify. And importantly the incompleteness theorems still hold in contexts where you don't really have a meaningful model theory to speak of.
