Let $X$ be a set, let $\wp(X)$ be the collection of all subsets of $X$, and let $\pi$ and $\sigma$ be any subcollections of $\wp(X)$. Consider the collection \begin{equation*} \{R\cap S:R\in\pi,S\in\sigma\}, \end{equation*} is there a standard notation for this collection?
My interest in this collection comes from the following. Let $\wp^2(X)$ denote the collection of all subsets of $\wp(X)$, and let $\lesssim$ denote the "reversed refinement order" on $\wp^2(X)$, i.e., $\pi\lesssim\sigma$ if and only if for each $R$ in $\pi$ there is a $S$ in $\sigma$ such that $R\subseteq S$, for any $\pi,\sigma\in\wp^2(X)$. Then $\lesssim$ is a preorder on $\wp^2(X)$. Let $\sim$ be the relation on $\wp^2(X)$ defined for any $\pi$ and $\sigma$ in $\wp^2(X)$ by $\pi\sim\sigma$ if and only if $\pi\lesssim\sigma$ and $\sigma\lesssim\pi$. Then $\sim$ is an equivalence relation on $\wp^2(X)$, and a partial order $\leq$ on the collection $\wp^2(X)/\sim$ of equivalence classes $[\pi]$ ($\pi\in\wp^2(X)$) can be defined by \begin{equation*} [\pi]\leq[\sigma]\quad\text{if and only if}\quad\pi\lesssim\sigma. \end{equation*} If $\upsilon$ is the collection $\{R\cap S:R\in\pi,S\in\sigma\}$, then the equivalence class $[\upsilon]$ would be the greatest lower bound of $[\pi]$ and $[\sigma]$ under the partial order $\leq$.
$$ \def\leq{\leqslant} \def\geq{\geqslant} \def\cP#1{{\cal P}(#1)} \def\PpS{{\cal P}^\uparrow(S)} $$ I don't know of a specific notation for this case, but as your operation is an instance of a more general construction, you might be interested to know about this more general setting.
If $S$ is a semigroup, the set $\cP{S}$ of subsets of $S$ is also a semigroup, for the product defined, for every $X, Y \in \cP{S}$, by $$ XY = \{ xy \mid x \in X, y \in Y \} $$ This semigroup is called the power semigroup (or global semigroup) of $S$. Thus your construction is just the power semigroup of the semigroup $S = (\cP{X}, \cap)$.
There is a similar notion for ordered semigroups. An ordered semigroup is a semigroup $S$ endowed with a partial order $\leq$ such that $x \leq y$ implies $zx \leq zy$ and $xz \leq yz$, that is, a stable preorder.
Let $(S,\leq)$ be an ordered semigroup. One can define a preorder relation $\leq$ on $\cP{S}$ by setting $X \leq Y$ if and only if, for each $y\in Y$, there exists $x \in X$ such that $x \leq y$. It is easy to see that this defines a stable preorder on $\cP{S}$. Denote by $\sim$ the equivalence relation defined by $X \sim Y$ if and only if $X \leq Y$ and $Y \leq X$. Then $\sim$ is a congruence on $\cP{S}$ and the preorder $\leq$ induces a stable partial order on the semigroup $\cP{S}/{\sim}$.
But there is a more direct way to obtain the resulting ordered semigroup. Let $\PpS$ be the set of upper sets of $(S, \leq)$. Define the product of two elements $U$ and $V$ of $\PpS$ as the upper set generated by $UV$ and take inclusion as the order on $\PpS$. I let you verify that you obtain the same ordered semigroup as above.
In the second part of your post, you apply this construction to the ordered semigroup $(S, \leq) = (\cP{X}, \cap, \subseteq)$, except that you reverse the order at the very end.