Question: Are there any references explaining the definition (or definitions) of quotient objects in the category of lattices?
In particular a characterization in terms of universal properties would be helpful for comparing with other notions of "quotient object". Multiple possible category-theoretic notions of quotient object exist that generalize the construction in $Set$, cf. Wikipedia and nLab, and I am curious about which apply to lattices.
Note: unlike other questions about "quotient lattices" on this website, I am asking about the order-theoretic notion of "lattice". I put some partial progress I've made as a community wiki "answer" below, but it only seems to apply to distributive lattices.
Motivation: Given a set $W$ and a fixed subset $U \subseteq W$, we have the sublattice $\mathscr{L}$ of the powerset lattice $2^W$ consisting of all $V$ such that $U \subseteq V \subseteq W$, $\mathscr{L}:= \{V \subseteq W: U \subseteq V\}$. We also have the sublattice $2^U$.
Heuristically at least, it seems like it should be true that "$\mathscr{L} \cong 2^W / 2^U$", but that of course would require $2^W / 2^U$ to be defined.
In analogy to quotients in vector spaces, where $v_1 \cong_{U} v_2 \iff v_1 - v_2 \in U$ for $U$ some "absorbing"/"annihilating"/"kernel"/"null" subspace, we can define an equivalence relation on $2^W$ by $V_1 \cong_U V_2 \iff V_1 \setminus V_2 \subseteq U \text{ and } V_2 \setminus V_1 \subseteq U$.
An equivalent definition of $V_1 \cong_U V_2$ is $V_1 = V_2 \cup \tilde{U}_1$ and $V_2 = V_1 \cup \tilde{U}_2$ for some $\tilde{U}_1, \tilde{U}_2 \subseteq U$, i.e. $\tilde{U}_1, \tilde{U}_2 \in 2^U$.
Note that an equivalent definition of $\mathscr{L}$ is $\{V \cup U: V \subseteq W\}$, and I claim that as a result there is a bijection between $\mathscr{L}$ and the quotient set $2^W / \cong_U$. So if one can define "$2^W / 2^U$" as $2^W / \cong_U$, then we would have a notion of "quotient lattice" making the heuristic notion "$\mathscr{L} \cong 2^W / 2^U$" both well-defined and true.
Partial progress: The equivalence relation defined above can be generalized to an arbitary join semi-lattice $L$.
Specifically, given a join-semilattice $L$, and a (for now) arbitrary subset $M \subseteq L$, we can define the equivalence relation $\cong_M$ on $L$ as follows: $y_1 \cong_M y_2 \iff \exists x_1, x_2 \in M \text{ such that }y_1 = y_2 \lor x_1 \text{ and }y_2 = y_1 \lor x_2$.
If we want the "projection" function from $L$ to $L / \cong_M$ to be a (semi)lattice homomorphism, and thus $L / \cong_M$ to be a well-defined "quotient object", then we need the relationship $[y] \lor [z] := [y \lor z]$ to be true and well-defined.
I believe I was able to show that this definition of join $\lor$ on $L / \cong_M$ is well-defined as long as $M$ itself is closed under $\lor$, e.g. when it is a join-semilattice. So at the very least there seems to be a well-defined notion of "quotient join-semilattice".
If additionally $L$ is a lattice, and we want the quotient set $L / \cong_M$ to be a lattice as well, then we need the projection function $L \to L / \cong_M$ to be a lattice homomorphism, so not just preserve joins as above but also preserve meets, i.e. the relationship $[y] \land [z] = [y \land z]$ also needs to be well-defined and true.
I believe I was able to show that this holds provided that (1) $L$ is not just an arbitary lattice but specifically a distributive lattice, and (2) $M$ is not just a join-semilattice (closed under "internal" joins) but also a down-set (a.k.a. order ideal).
In particular, for any $\ell_1, \ell_2 \in L$, we have $\ell_1 \land \ell_2 \le \ell_1$, i.e. meets are always decreasing, so $M$ is a down-set ($m \in M$, $\ell \le m$ $\implies \ell \in M$) if and only if $\ell \land m \in M$ for all $m \in M$ and all $\ell \in L$. I think this is because of the lattice absorption identity, $\ell \le m \implies \ell \land m = \ell$. Notice that being a down-set implies being closed not just under all "internal meets" (which would just be a meet-semilattice), $m_1 \land m_2 \in M$ for all $m_1, m_2 \in M$, but also under all "external meets". The analogy with the ring-theoretic notion of "ideal" ("closure under external multiplication") is presumably the source of the term "order ideal".
Anyway, in conclusion it seems like whenever (1) $L$ is a distributive lattice, and (2) $M$ is a sublattice (join-semilattice and meet-semilattice) that is also a down-set, then $L / \cong_M$ is also a lattice, and so we have a well-defined and meaningful notion of "quotient lattice" $L / M$. (Notice that $2^W$ and $2^U$ in the motivating example above satisfy these conditions.)
Presumably $M$ and $L/M$ inherit the distributive identities from $L$ (via restriction and the projection lattice homomorphism respectively), so more specifically it seems that we have defined "quotient objects" in the (sub)category of distributive lattices whenever the "divisor lattice" is additionally an order ideal.
This seems analogous to the situation for rings, where quotients are not necessarily well-defined when the divisors are arbitrary subrings, but quotients are well-defined when the divisors are ideals. And it might not just be an analogy, to the extent that distributive lattices can be identified with or embedded inside of Booelan lattices that are in turn equivalent to Boolean rings?