A cardinal κ is a Berkeley cardinal, if for any transitive set $M$ with $κ∈M$ and any ordinal $α<κ$ there is an elementary embedding $j : M → M$ with $\alpha<\text{crit}(j)<\kappa$.

My question is about the restrictions involved in that definition of $\kappa \in M$? and of $M$ being *transitive*? and of $M$ being a *set*? why not the following?

**....if for any class $M$ with $κ \subseteq M$ and any ordinal $α<κ$ there is an elementary embedding $j : M → M$ with $\alpha<\text{crit}(j)<\kappa$.**

Where the underlying theory is $MK$ like, with the axiom of limitation of size replaced by an axiom of limitation of size on sets, more specifically any class that is subnumerous to a set, is a set. And of course without $AC$.