An important feature which separates the notion of virtual large cardinals from the related notion of generic large cardinals is that we only consider embeddings on set-sized structures. Since most large cardinals are characterized by embeddings of the entire universe, there will always be some arbitrary choices made in the characterization we pick because we need to reduce to a set-sized embedding.

It is true that Schindler originally thought of remarkable cardinals as virtual supercompacts using the Magidor characterization. But in fact, the most natural characterization of supercompact cardinals in terms of set-sized embedding: $\kappa$ is supercompact if for every $\lambda>\kappa$, there is $\alpha>\lambda$ and a transitive model $N$ closed under $\lambda$-sequences with an elementary embedding $j:V_\alpha\to N$ with critical point $\kappa$, when virtualized gives a remarkable cardinal! (Note that $N$ is assumed to be closed under $\lambda$-sequences in $V$.) So remarkable cardinals is not an example you are looking for. Vopenka's Principle comes closer.

Bagaria showed that Vopenka's Principle is equivalent to the existence of a $C^{(n)}$-extendible cardinal for every $1\leq n<\omega$. $C^{(n)}$ is the collection of all cardinal $\alpha$ such that $V_\alpha\prec_{\Sigma_n} V$. A cardinal $\kappa$ is $C^{(n)}$-extendible if for every $\alpha\in C^{(n)}$ above $\kappa$, there is an elementary $j:V_\alpha\to V_\beta$ with critical point $\kappa$ and $\beta\in C^{(n)}$. It is not difficult to see that we can always obtain a $C^{(n)}$-extendibility embedding with the additional property that $j(\kappa)>\alpha$. The proof makes use of the Kunen inconsistency and it turns out that this is fundamental.

Kunen's inconsistency does not hold for virtual embeddings in the sense that in a forcing extension we can have embeddings $j:V_\lambda\to V_\lambda$ with $\lambda$ much greater than the supreumum of the critical sequence of $j$. This has the effect of messing up some properties of virtual $C^{(n)}$-extendible cardinals and virtual Vopenka's Principle. We showed with Joel Hamkins that virtual $C^{(n)}$-extendible cardinals with the assumption that $j(\kappa)>\alpha$ are not equivalent to virtual $C^{(n)}$-extendible cardinals, call them virtual weakly $C^{(n)}$-extendible cardinals, without this assumption (see here). However the two notions are equiconsistent, so the consistency strength is not affected by the different characterizations.

The consequence for virtual Vopenka's Principle is that it is no longer equivalent to the existence of a virtual $C^{(n)}$-extendible cardinal for every $1\leq n<\omega$, but to the existence of the virtual weak $C^{(n)}$-extendibles.

It seems that the only instance of equivalent characterizations producing different virtual notions all revolve around the consequences of the failure of the Kunen Inconsistency.

In practice the most robust embedding characterizations for virtualizing have the form $j:V_\alpha\to V_\beta$, namely the target model has the form $V_\beta$. In my opinion, the reason that it appears that strong cardinals and supercompact cardinals have equivalent virtual versions is that strong cardinals simply don't have a virtual characterization because they don't have a "robust" embedding characterization for virtualizing.

An important feature which separates the notion of virtual large cardinals from the related notion of generic large cardinals is that we only consider embeddings on set-sized structures. Since most large cardinals are characterized by embeddings of the entire universe, there will always be some arbitrary choices made in the characterization we pick because we need to reduce to a set-sized embedding.

It is true that Schindler originally thought of remarkable cardinals as virtual supercompacts using the Magidor characterization. But in fact, the most natural characterization of supercompact cardinals in terms of set-sized embedding: $\kappa$ is supercompact if for every $\lambda>\kappa$, there is $\alpha>\lambda$ and a transitive model $N$ closed under $\lambda$-sequences with an elementary embedding $j:V_\alpha\to N$ with critical point $\kappa$, when virtualized gives a remarkable cardinal! (Note that $N$ is assumed to be closed under $\lambda$-sequences in $V$.) So remarkable cardinals is not an example you are looking for. Vopenka's Principle comes closer.

Bagaria showed that Vopenka's Principle is equivalent to the existence of a $C^{(n)}$-extendible cardinal for every $1\leq n<\omega$. $C^{(n)}$ is the collection of all cardinal $\alpha$ such that $V_\alpha\prec_{\Sigma_n} V$. A cardinal $\kappa$ is $C^{(n)}$-extendible if for every $\alpha\in C^{(n)}$ above $\kappa$, there is an elementary $j:V_\alpha\to V_\beta$ with critical point $\kappa$ and $\beta\in C^{(n)}$. It is not difficult to see that we can always obtain a $C^{(n)}$-extendibility embedding with the additional property that $j(\kappa)>\alpha$. The proof makes use of the Kunen inconsistency and it turns out that this is fundamental.

Kunen's inconsistency does not hold for virtual embeddings in the sense that in a forcing extension we can have embeddings $j:V_\lambda\to V_\lambda$ with $\lambda$ much greater than the supreumum of the critical sequence of $j$. This has the effect of messing up some properties of virtual $C^{(n)}$-extendible cardinals and virtual Vopenka's Principle. We showed with Joel Hamkins that virtual $C^{(n)}$-extendible cardinals with the assumption that $j(\kappa)>\alpha$ are not equivalent to virtual $C^{(n)}$-extendible cardinals, call them virtual weakly $C^{(n)}$-extendible cardinals, without this assumption (see here). However the two notions are equiconsistent, so the consistency strength is not affected by the different characterizations.

The consequence for virtual Vopenka's Principle is that it is no longer equivalent to the existence of a virtual $C^{(n)}$-extendible cardinal for every $1\leq n<\omega$, but to the existence of the virtual weak $C^{(n)}$-extendibles.

It seems that the only instance of equivalent characterizations producing different virtual notions all revolve around the consequences of the failure of the Kunen Inconsistency.

In practice the most robust embedding characterizations for virtualizing have the form $j:V_\alpha\to V_\beta$, namely the target model has the form $V_\beta$. In my opinion, the reason that it appears that strong cardinals and supercompact cardinals have equivalent virtual versions is that strong cardinals simply don't have a virtual characterization because they don't have a "robust" embedding characterization for virtualizing. A discussion of this can be found in our joint paper with Ralf Schindler (see here).