Given a logic (= regular logic in the sense of Ebbinghaus/Flum/Thomas) $\mathcal{L}$, let a $\downarrow$-sentence be an $\mathcal{L}$-sentence $\varphi$ such that, whenever $\mathfrak{M}\models\varphi$ and $\mathfrak{N}$ is a substructure of $\mathfrak{M}$, we have $\mathfrak{N}\models\varphi$ as well. For instance, the proof of the Łoś–Tarski theorem shows that every $\downarrow$-sentence in a compact extension of $\mathsf{FOL}$ is equivalent to an $\mathsf{FOL}$-sentence of $\forall^*$-form.
Now say that $\mathcal{L}$ is internally-dLS iff for each language $\Sigma$ and each $\mathcal{L}[\Sigma]$-sentence $\varphi$ there is a larger language $\Pi$ and an $\mathcal{L}[\Pi]$-sentence $\psi$ such that
$\psi$ is a $\downarrow$-sentence, and
for every $\Sigma$-structure $\mathfrak{A}$ we have $\mathfrak{A}\models\varphi$ iff $\mathfrak{A}$ has an expansion satisfying $\psi$.
This is a very strong form of the downward (to $\vert\mathcal{L}\vert$) Löwenheim–Skolem property, but it's also what the usual proof of the downward Löwenheim–Skolem theorem for first-order logic actually establishes (the existence of $\psi$ corresponding to Skolemization. I'm curious how common this coincidence is:
Are there "natural" logics which have the downward Löwenheim–Skolem property but are not internally-dLS?
(Hopefully this isn't too vague!)
One reason I'm interested in this is that the Skolem property leads to an easy proof of a weak form of Lindström's theorem: if $\mathcal{L}$ is an internally-dLS compact logic extending $\mathsf{FOL}$, then we can use the argument of the Los-Tarski theorem to show that the $\mathcal{L}$-elementary classes are all $\mathsf{FOL}$-pseudoelementary, and the result then follows via Craig interpolation (for $\mathsf{FOL}$, not $\mathcal{L}$). Not only does this avoid the characterization of $\equiv_{\mathsf{FOL}}$ via EF-games, it also doesn't use the hypothesis that $\mathcal{L}$ has the relativization property, which is the most difficult aspect of the definition of regular logics.
