= Solution
Let $\lambda$ be inaccessible and let $j:V_\lambda\to M$ be an elementary embedding into a transitive set with critical point $\kappa$. It is <beta-strong elementary embedding>[$\beta$-strong] when
$$
V_{\kappa+\beta}\subseteq M.
$$
A formula $\Phi(x,\kappa)$ is a <beta-stable cardinal property> when it is absolute between the universe and every transitive set containing $V_{\kappa+\beta}$.
To say that the embedding reflects such a property means that whenever $\Phi(\kappa)$ holds,
$$
\{\mu<\kappa:\Phi(\mu)\}
$$
is unbounded in $\kappa$: for every $\gamma<\kappa$ there is such a $\mu$ with $\gamma<\mu<\kappa$. Indeed, stability gives $M\models\Phi(\kappa)$, so $M$ sees the witness $\kappa$ between $\gamma$ and $j(\kappa)$. Elementarity reflects a witness between $\gamma$ and $\kappa$. This is <reflection by a beta-strong embedding>.
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