According to Wikipedia, a theory $T$ that interprets arithmetic is consistent with the $\omega$-rule if and only if it has an $\omega$-model. That would mean that consistency with the $\omega$-rule is not absolute to a minimal $\omega$-model of $T$ (that is, a model of $T+\text{"there is no $\omega$-model of T"}$; such a model exist if $\omega$-models exist).
However, it appears that consistency with the $\omega$-rule is expressible as a schema of arithmetical sentences. We can arithmetize provability by at most $n$ applications of the $\omega$-rule as $Pr^{\omega,n}_T(\ulcorner \phi \urcorner)$ (we use Quine quotes to denote Gödel numbers of formulas), defined recursively by $Pr^{\omega,0}_T(\ulcorner \phi \urcorner) \equiv Pr_T(\ulcorner \phi \urcorner)$ (the arithmetized provability relation), $$Pr^{\omega,n+1}_T(\ulcorner \forall n \phi(n) \urcorner) \equiv \forall n \exists \ulcorner \psi \urcorner (\text{$\psi$ is has one free variable} \wedge Pr_T(\ulcorner \forall m \in \mathbb{N} \psi \Rightarrow \phi(n) \urcorner) \wedge Pr^{\omega,n}_T(\ulcorner \forall m \psi \urcorner)$$ and $Pr^{\omega,n}_T(\ulcorner \exists n \phi(n) \urcorner) \equiv \exists n Pr^{\omega,n}_T(\ulcorner \phi(n) \urcorner)$ From this definition we can arithmetize consistency with the $\omega$-rule as the schema $\forall \ulcorner \phi \urcorner Pr^{\omega,n}_T(\ulcorner \phi \urcorner) \Rightarrow \neg Pr_T(\ulcorner \neg \phi \urcorner)$, where $n$ ranges over the natural numbers in the metatheory. Since this is schema of arithmetical sentences, it should be absolute to any $\omega$-model of second-order arithmetic with a truth predicate for arithmetical truth.
Where does this reasoning go wrong?