Vopenka's Principle ($VP$) states that, given any proper class $\mathcal{C}$ of structures in the same (set-sized, relational) signature $\Sigma$, there are some distinct $A, B\in\mathcal{C}$ such that $A$ is isomorphic to an elementary substructure of $B$. In terms of consistency, we have the following rough upper and lower bounds: $$\text{proper class of extendibles $\le$ Vopenka's Principle $\le$ almost huge.} $$ (I don't know if this is state-of-the-art; more precise bounds, if known, would be welcome!) Thus, even though on the face of it $VP$ does not directly talk about cardinals, it is generally thought of as a large cardinal axiom.

Now, abstract model theory appears to give a framework for generalizing VP. Let $\mathcal{L}$ be any regular logic$^*$; then we can study "Vopenka's Principle for $\mathcal{L}$," $VP(\mathcal{L})\equiv$ "For any proper class $\mathcal{C}$ of $\Sigma$-structures ($\Sigma$ a set-sized relational signature), there are distinct $A, B\in\mathcal{C}$ with $A$ $\mathcal{L}$-elementarily embeddable into $B$." So, for example, taking $\mathcal{L}_I$ to denote first-order logic, $VP$ is just $VP(\mathcal{L}_I)$.

Stronger Versions: Two reasonable logics to look at for stronger versions of $VP$ are $\mathcal{L}_{II}$ and $\mathcal{L}_{\omega_1\omega}$, second-order and (the smallest standard) infinitary logic respectively. However, the corresponding Vopenka principles are still just as strong as $VP(\mathcal{L}_I)$.$^{**}$ In general, $VP(\mathcal{L}_I)$ seems to be an upper bound for Vopenka's Principles for locally set-sized, definable logics. Since non-definable logics are of limited interest, it's reasonable to look at class-sized logics. The tamest class-sized logic I know of is $\mathcal{L}_{\infty\omega}$, the infinitary logic allowing arbitrary set-sized Boolean combinations but no infinite strings of quantifiers. However, $VP(\mathcal{L}_{\infty\omega})$ is inconsistent: by a famous theorem of Carol Karp, two structures are $\mathcal{L}_{\infty\omega}$-equivalent if and only if they are back-and-forth equivalent, so the class $\mathcal{O}$ of all ordinals (regarded as linear orderings) is a counterexample in any model of $ZFC$.

This all suggests that there are probably no interesting versions of Vopenka's Principle stronger than the usual one, and that any weaker form of Vopenka has to come from a horribly weak - to the point of being probably uninteresting - logic. I find this kind of disappointing. So, my question is:

Are there any interesting logics $\mathcal{L}$ for which $VP(\mathcal{L})$ is different from the usual Vopenka's Principle?

Vopěnka's Principle ($VP$) states that, given any proper class $\mathcal{C}$ of structures in the same (set-sized, relational) signature $\Sigma$, there are some distinct $A, B\in\mathcal{C}$ such that $A$ is isomorphic to an elementary substructure of $B$. In terms of consistency, we have the following rough upper and lower bounds: $$\text{proper class of extendibles $\le$ Vopěnka's Principle $\le$ almost huge.} $$ (I don't know if this is state-of-the-art; more precise bounds, if known, would be welcome!) Thus, even though on the face of it $VP$ does not directly talk about cardinals, it is generally thought of as a large cardinal axiom.

Now, abstract model theory appears to give a framework for generalizing VP. Let $\mathcal{L}$ be any regular logic$^*$; then we can study "Vopěnka's Principle for $\mathcal{L}$," $VP(\mathcal{L})\equiv$ "For any proper class $\mathcal{C}$ of $\Sigma$-structures ($\Sigma$ a set-sized relational signature), there are distinct $A, B\in\mathcal{C}$ with $A$ $\mathcal{L}$-elementarily embeddable into $B$." So, for example, taking $\mathcal{L}_I$ to denote first-order logic, $VP$ is just $VP(\mathcal{L}_I)$.

Stronger Versions: Two reasonable logics to look at for stronger versions of $VP$ are $\mathcal{L}_{II}$ and $\mathcal{L}_{\omega_1\omega}$, second-order and (the smallest standard) infinitary logic respectively. However, the corresponding Vopěnka principles are still just as strong as $VP(\mathcal{L}_I)$.$^{**}$ In general, $VP(\mathcal{L}_I)$ seems to be an upper bound for Vopěnka's Principles for locally set-sized, definable logics. Since non-definable logics are of limited interest, it's reasonable to look at class-sized logics. The tamest class-sized logic I know of is $\mathcal{L}_{\infty\omega}$, the infinitary logic allowing arbitrary set-sized Boolean combinations but no infinite strings of quantifiers. However, $VP(\mathcal{L}_{\infty\omega})$ is inconsistent: by a famous theorem of Carol Karp, two structures are $\mathcal{L}_{\infty\omega}$-equivalent if and only if they are back-and-forth equivalent, so the class $\mathcal{O}$ of all ordinals (regarded as linear orderings) is a counterexample in any model of $ZFC$.

This all suggests that there are probably no interesting versions of Vopěnka's Principle stronger than the usual one, and that any weaker form of Vopěnka has to come from a horribly weak - to the point of being probably uninteresting - logic. I find this kind of disappointing. So, my question is:

Are there any interesting logics $\mathcal{L}$ for which $VP(\mathcal{L})$ is different from the usual Vopěnka's Principle?

Stronger Versions: Two reasonable logics to look at for stronger versions of $VP$ are $\mathcal{L}_{II}$ and $\mathcal{L}_{\omega_1\omega}$, second-order and (the smallest standard) infinitary logic respectively. However, the corresponding Vopenka principles are still just as strong as $VP(\mathcal{L}_I)$.$^{**}$ In general, $VP(\mathcal{L}_I)$ seems to be an upper bound for Vopenka's Principles for locally set-sized, definable logics. Since non-definable logics are of limited interest, it's reasonable to look at class-sized logics. The tamest class-sized logic I know of is $\mathcal{L}_{\infty\omega}$, the infinitary logic allowing arbitrary set-sized Boolean combinations but no infinite strings of quantifiers. However, $VP(\mathcal{L}_{\infty\omega})$ is inconsistent: by a famous theorem of Carol Karp, two structures are $\mathcal{L}_{\infty\omega}$-equivalent if and only if they are back-and-forth equivalent, so the class $\mathcal{O}$ of all ordinals (as linear orderings) is a counterexample in any model of $ZFC$.

Stronger Versions: Two reasonable logics to look at for stronger versions of $VP$ are $\mathcal{L}_{II}$ and $\mathcal{L}_{\omega_1\omega}$, second-order and (the smallest standard) infinitary logic respectively. However, the corresponding Vopenka principles are still just as strong as $VP(\mathcal{L}_I)$.$^{**}$ In general, $VP(\mathcal{L}_I)$ seems to be an upper bound for Vopenka's Principles for locally set-sized, definable logics. Since non-definable logics are of limited interest, it's reasonable to look at class-sized logics. The tamest class-sized logic I know of is $\mathcal{L}_{\infty\omega}$, the infinitary logic allowing arbitrary set-sized Boolean combinations but no infinite strings of quantifiers. However, $VP(\mathcal{L}_{\infty\omega})$ is inconsistent: by a famous theorem of Carol Karp, two structures are $\mathcal{L}_{\infty\omega}$-equivalent if and only if they are back-and-forth equivalent, so the class $\mathcal{O}$ of all ordinals (regarded as linear orderings) is a counterexample in any model of $ZFC$.