Bumped, since I was recently thinking about these again.
My friend introduced the following notion:
Let $\xi > 0$, $\eta$ be ordinals, $n$ be a natural number and $\mathcal{A}, X$ be classes. A cardinal $\kappa$ is called $\mathcal{A}\textrm{-}\eta\textrm{-}(n,\xi)$-weird on $X$ iff there is a $\Sigma^1_{2n}$-elementary embedding $j: \langle V_{\kappa+\eta}; \in, \mathcal{A} \cap V_{\kappa+\eta} \rangle \to \langle V_{\kappa+\eta+\xi}; \in, \mathcal{A} \cap V_{\kappa+\eta+\xi} \rangle $ with critical point $\kappa$ and $j(\kappa) \in X$.
How high in consistency strength is it?
