Elementary embedding between infinite complete atomic Boolean algebras

Question

This question is related to my previous question: Elementary embedding between infinite atomic Boolean algebras.

Let $S$ be an infinite set of atoms and let $At_{comp}(S)$ be the complete atomic Boolean algebra generated by $S$ (the powerset Boolean algebra on $S$).

Now suppose we expand $S$ to $S'$ by adding some other atoms, and let $At_{comp}(S')$ be the complete atomic Boolean algebra generated by $S'$.

Let $\mathscr{L}$ be the language of Boolean algebra and $T$ be the theory of infinite complete atomic Boolean algebra in $\mathscr{L}$. $At_{comp}(S)$ and $At_{comp}(S')$, as models of $\mathscr{L}$, are both models of $T$ (and hence they are elementarily equivalent). Now my question is: is $At_{comp}(S)$ an elementary submodel of $At_{comp}(S')$, in the sense that for any $\phi(x_1, ..., x_n)$ of $\mathscr{L}$, and $a_1, ..., a_n \in At_{comp}(S)$, $$At_{comp}(S) \models \phi(x_1, ..., x_n)[a_1, ..., a_n] \, \, \text{iff}\, \, At_{comp}(S') \models \phi(x_1, ..., x_n)[a_1, ..., a_n]$$

I also wonder about a varient of the above question: let $At_{comp}(S)$ be the complete atomic Boolean algebra generated by $S$. Let $B$ be some infinite atomic Boolean algebra such that:

1. $S'$ is the set of atoms in $B$.
2. For every subset $D \subseteq S$, $\bigcup{D}$ and $\bigcap{D}$ are in $B$. That is, $At_{comp}(S) \subseteq B$.
3. $B \models T$.

Is $At_{comp}(S)$ an elementary submodel of $B$?