I just read a proof and, after struggling some time with a mental leap, I think that it uses tacitly the following:

Let $\kappa$ be a regular cardinal, $\theta > \kappa$ a regular cardinal too then: $ S \subset \kappa$ is stationary if and only if $\forall \mathcal{A} = (H(\theta), \in, <,..) \exists M \prec \mathcal{A}, |M| < \kappa,$ such that $sup(M \cap \kappa) \in S$.

Now my questions are:

Is this statement above even true? (I think so as I have a proof, but this doesn't have to mean anything)

It appears to me that the latter part of this characterization is a quite strong assumption as $\mathcal{A}$ might contain a lot of additional information, so is there a possibility to weaken it? Or could you mention any similar statements to the one above?

Thank you

EDIT: I accepted the answer of Philip, simply because he has lower points. Francois answer would have deserved it too.