Saying "$True(g(A))\iff A$ holds" is implicitly saying that that statement is true in the standard model - in the same way that if I say "Fermat's last theorem is true," I mean that it is true in the standard model. So in fact the two versions are equivalent.
Indeed, if you want to make the model explicit, you should apply that to the $A$-side too: "$True(g(A))$ is true in the standard model iff $A$ is true in the standard model."
However, there's a value to phrasing it without reference to a specific model: Tarski's undefinability theorem doesn't only apply to the standard model! If $N$ is a model of $PA$, then there is no formula $True$ such that $N\models True(g(A))\iff N\models A$, via the same proof as usual, the key point being that the conclusion of the diagonal lemma is about provability in $PA$, not merely truth in the standard model.
EDIT: I think it would help if Tarski's theorem were stated in the following way (and indeed, this is how I state it when I teach it):
If $A\models PA$, then there is no formula $\varphi(x)$ (in one free variable, in the language of arithmetic) such that for all sentences $p$ in the language of arithmetic, $$A\models \varphi(g(p))\quad\iff\quad A\models p,$$ where "$g(\cdot)$" is the Goedel number function.