This is an attempt to locate a theorem I vaguely remember but cannot find (which arose in the context of Consistency of non-trivial elementary embedding $j : \mathit{Ord} \to \mathit{Ord}$ and motivated Can $Ord$ have nontrivial second-order elementary self-embeddings?). Specifically, I recall a theorem along the following lines (in contrast of course with the Kunen inconsistency):
Every "reasonably definable" expansion of the class-sized structure $(\mathit{Ord}; \in)$ has nontrivial elementary self-embeddings.
Here the relevant notion of "reasonably definable" was broad enough to include the usual ordinal arithmetic operations, and if I recall correctly more complicated operations like $\alpha\mapsto\omega_\alpha$. On the other hand, it was a $\mathsf{ZFC}$ theorem, so we certainly can't go all the way to second-order logic (see the MO question linked above).
Additionally, I vaguely recall that this was due to either Harvey Friedman or Mostowski and that the relevant definability notion was "least-fixed-point-flavored," but I'm much less confident about either of these memories.
\mathiton your $\mathit{Ord}$s. I also inlined the titles of the questions you reference. I hope that this is all right. $\endgroup$