Below, by "logic" I mean "regular logic without equality;" see Badia/Caicedo/Noguera, Maximality of logic without identity.

Given a logic $\mathcal{L}$, a structure $\mathfrak{M}$, and an equivalence relation $E$ on $\mathfrak{M}$, say that $E$ is $\mathcal{L}$-equality-like iff for every $\mathcal{L}$-formula $\varphi(\overline{x})$ with parameters from $\mathfrak{M}$ and every pair of appropriate-length tuples of elements $\overline{a},\overline{b}$ of $\mathfrak{M}$ with $a_iEb_i$ for each $i$, we have $\varphi(\overline{a})^\mathfrak{M}\iff\varphi(\overline{b})^\mathfrak{M}$. Basically, $E$ is $\mathcal{L}$-equality-like iff it satisfies the substitution principle for $\mathcal{L}$-formulas with parameters.

Now an $\mathcal{L}$-equality-like relation is automatically a congruence, so we can consider quotients of structures by $\mathcal{L}$-equality-like relations. Say that a logic $\mathcal{L}$ is simple-for-equality iff $$(\mathfrak{M}, E)\equiv_\mathcal{L}(\mathfrak{M}/E,=)$$ for every structure $\mathfrak{M}$ and every $\mathcal{L}$-equality-like $E$ on $\mathfrak{M}$. Intuitively, $\mathcal{L}$ is simple-for-equality if the entire behavior of equality, as far as $\mathcal{L}$ is concerned, is captured by (the equivalence relation axioms and) the substutitution principle.

It is that equality-free first-order logic $\mathsf{FOL_{w/o=}}$ is simple-for-equality, while its expansion $\mathsf{FOL_{w/o=}}(\exists!)$ by the unique-existence quantifier is not simple-for-equality (see this earlier answer of mine). However, this is rather contrived since in some sense equality is already implicit in $\exists!$. My first question is whether there is a better example:

Question 1: Are there any "natural" logics which are not simple-for-equality?

My second question, in the opposite direction, is whether there are any more-standard 'tameness' properties of logics which entail simplicity-for-equality:

Question 2: Are there already-studied model-theoretic properties which imply that a logic is simple-for-equality (and so could explain a lack of natural examples)?

Below, by "logic" I mean "regular logic without equality;" see Badia/Caicedo/Noguera, Maximality of logic without identity.

Given a logic $\mathcal{L}$, a structure $\mathfrak{M}$, and an equivalence relation $E$ on $\mathfrak{M}$, say that $E$ is $\mathcal{L}$-equality-like iff for every $\mathcal{L}$-formula $\varphi(\overline{x})$ with parameters from $\mathfrak{M}$ and every pair of appropriate-length tuples of elements $\overline{a},\overline{b}$ of $\mathfrak{M}$ with $a_iEb_i$ for each $i$, we have $\varphi(\overline{a})^\mathfrak{M}\iff\varphi(\overline{b})^\mathfrak{M}$. Basically, $E$ is $\mathcal{L}$-equality-like iff it satisfies the substitution principle for $\mathcal{L}$-formulas with parameters.

Now an $\mathcal{L}$-equality-like relation is automatically a congruence, so we can consider quotients of structures by $\mathcal{L}$-equality-like relations. Say that a logic $\mathcal{L}$ is simple-for-equality iff $$(\mathfrak{M}, E)\equiv_\mathcal{L}(\mathfrak{M}/E,=)$$ for every structure $\mathfrak{M}$ and every $\mathcal{L}$-equality-like $E$ on $\mathfrak{M}$. Intuitively, $\mathcal{L}$ is simple-for-equality if the entire behavior of equality, as far as $\mathcal{L}$ is concerned, is captured by (the equivalence relation axioms and) the substutitution principle.

It's easy to show that, while both full first-order logic and its equality-free version are simple-for-equality, the intermediate logic gotten from the latter by adding the unique-existence quantifier is not (see this earlier answer of mine). However, this is rather contrived since in some sense equality is already implicit in $\exists!$. My first question is whether there is a better example:

Question 1: Are there any "natural" logics which are not simple-for-equality?

My second question, in the opposite direction, is whether there are any more-standard 'tameness' properties of logics which entail simplicity-for-equality:

Question 2: Are there already-studied model-theoretic properties which imply that a logic is simple-for-equality (and so could explain a lack of natural examples)?