This MathOverflow question by Trevor Wilson defines weakly Shelah cardinals as follows:

A cardinal $\kappa$ is weakly Shelah if for all $f : \kappa \to \kappa$ there is some $\alpha < \kappa$ that is closed under $f$ and there is some elementary embedding $j : V \to M$ (where $M$ is a transitive class) such that $\operatorname{crit}(j) = \alpha$ and $j(\alpha) > \kappa$ and $V_{j(f)(\kappa)} \subset M$.

I would like to add a requirement that $j(f) \upharpoonright \kappa = f$ but as a comment by Sean Cox on this question made me realize, it is not clear that that definition is equivalent to Trevor Wilson's definition. Are the definitions equivalent?