= Solution
There is a <bounded definition of omega>. Transitivity has the bounded definition $\operatorname{Trans}(x)\equiv\forall u\in x\,\forall v\in u\,(v\in x)$. An <ordinal> is a <transitive set> of <transitive sets> on which membership is a strict total order; this too has a <bounded formula in set theory>, since <ZF> supplies well-foundedness by the <Axiom of foundation>.
Define the bounded predicate
$$
L(x)\equiv\operatorname{Ord}(x)\land x\ne\varnothing\land\forall y\in x\,\exists z\in x\,(y\in z).
$$
For an <ordinal>, this says that it is a nonzero <limit ordinal>. It follows in <ZF> that
$$
\boxed{x=\omega\iff L(x)\land\forall y\in x\,\neg L(y),}
$$
since $\omega$ is the least nonzero <limit ordinal>. The right side is a <bounded formula in set theory>; consequently it is absolute between the specified <transitive models>. This also shows directly that those models have the same $\omega$.
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