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• $\mathsf{FOL}^{(\infty)}$ is (always up to expressive strength) a sublogic of $\mathcal{L}_{\infty,\omega}^L=\mathcal{L}_{\infty,\omega}\cap L$. More generally, if $M$ is an inner model and $\mathcal{L}$ is a set-sized sublogic of $\mathcal{L}_{\infty,\omega}^M$ which is appropriately definable in $M$, then $\mathcal{L}'\subseteq\mathcal{L}^M_{\infty,\omega}$ as well.

• $\mathsf{FOL}^{(\infty)}$ is strictly weaker than $\mathcal{L}_{\infty,\omega}^L$: if $(r_i)_{i\in\omega}$ are sufficiently mutually Cohen generic (say, generic over a countable elementary submodel of $L_{\omega_1}$), then the $\{<,\in\}$-structures $\omega\sqcup\{r_{2i}:i\in\omega\}$ and $\omega\sqcup\{r_{2i+1}: i\in\omega\}$ are $\mathsf{FOL}^{(\infty)}$-equivalent but not $\mathcal{L}_{\infty,\omega}^L$-equivalent.

• The argument of the previous bulletpoint implies that the $\mathsf{FOL}^{(\infty)}$-definable relations on $(\omega;<)$ fall well short of the constructible ones. On the other hand, it's also easy to see that they reach well past the hyperarithmetic: the $\mathsf{FOL}^{(\omega_1^{CK})}$-definable relations are exactly the hyperarithmetic ones, and $\mathcal{O}$ is then $\mathsf{FOL}^{(\omega_1^{CK}+1)}$-definable since a computable linear order is a well-order iff it supports a hyperarithmetic jump sequence.

• Finally, despite its clear weaknesses, for every structure $\mathfrak{A}$ the automorphism orbit relation on elements (or $n$-tuples for fixed $n$) of $\mathfrak{A}$ is $\mathsf{FOL}^{(\infty)}$-definable (this is just Scott's argument).

• $\mathsf{FOL}^{(\infty)}$ is (always up to expressive strength) a sublogic of $\mathcal{L}_{\infty,\omega}^L=\mathcal{L}_{\infty,\omega}\cap L$. More generally, if $M$ is an inner model and $\mathcal{L}$ is a set-sized sublogic of $\mathcal{L}_{\infty,\omega}^M$ which is appropriately definable in $M$, then $\mathcal{L}'\subseteq\mathcal{L}^M_{\infty,\omega}$ as well.

• $\mathsf{FOL}^{(\infty)}$ is strictly weaker than $\mathcal{L}_{\infty,\omega}^L$: if $(r_i)_{i\in\omega}$ are sufficiently mutually Cohen generic (say, generic over a countable elementary submodel of $L_{\omega_1}$), then the $\{<,\in\}$-structures $\omega\sqcup\{r_{2i}:i\in\omega\}$ and $\omega\sqcup\{r_{2i+1}: i\in\omega\}$ are $\mathsf{FOL}^{(\infty)}$-equivalent but not $\mathcal{L}_{\infty,\omega}^L$-equivalent.

• The argument of the previous bulletpoint implies that the $\mathsf{FOL}^{(\infty)}$-definable relations on $(\omega;<)$ fall well short of the constructible ones. On the other hand, it's also easy to see that they reach well past the hyperarithmetic: the $\mathsf{FOL}^{(\omega_1^{CK})}$-definable relations are exactly the hyperarithmetic ones, and $\mathcal{O}$ is then $\mathsf{FOL}^{(\omega_1^{CK}+1)}$-definable since a computable linear order is a well-order iff it supports a hyperarithmetic jump sequence.

• Finally, despite its clear weaknesses, for every countable structure $\mathfrak{A}$ the automorphism orbit relation on elements (or $n$-tuples for fixed $n$) of $\mathfrak{A}$ is $\mathsf{FOL}^{(\infty)}$-definable (this is just Scott's argument).

My guess is that these are exactly the relations in $L_\theta$, where $\theta$ is the smallest "gap ordinal" (= such that $L_\theta\cap\mathbb{R}=L_{\theta+1}\cap\mathbb{R}$). This $\theta$ is known to also be the least ordinal such that $L_\theta\models\mathsf{ZFC^-}$, or such that $L_\theta\cap\mathbb{R}\models\mathsf{Z}_2$. See this summary by Madore (entry 2.17).

My guess is that these are exactly the relations in $L_{\beta_0}$, where ${\beta_0}$ is the smallest "gap ordinal" (= such that $L_{\beta_0}\cap\mathbb{R}=L_{\beta_0+1}\cap\mathbb{R}$). This ${\beta_0}$ is known to also be the least ordinal such that $L_{\beta_0}\models\mathsf{ZFC^-}$, or such that $L_{\beta_0}\cap\mathbb{R}\models\mathsf{Z}_2$. See Putnam, Marek/Srebrny, or Madore (entry 2.17).

My guess is that these are exactly the relations in $L_\theta$, where $\theta$ is the smallest "gap ordinal" (= such that $L_\theta\cap\mathbb{R}=L_{\theta+1}\cap\mathbb{R}$).

My guess is that these are exactly the relations in $L_\theta$, where $\theta$ is the smallest "gap ordinal" (= such that $L_\theta\cap\mathbb{R}=L_{\theta+1}\cap\mathbb{R}$). This $\theta$ is known to also be the least ordinal such that $L_\theta\models\mathsf{ZFC^-}$, or such that $L_\theta\cap\mathbb{R}\models\mathsf{Z}_2$. See this summary by Madore (entry 2.17).