What is the definition of an intersection of sets of subsets?
Say, I have a set $X$ and its subsets $A,B$. I form 2 sets of subsets:
$X_1=\{X,\emptyset,A,X\setminus A\}\\ X_2=\{X,\emptyset,B,X\setminus B\}$,
what is $X_1\cap X_2$? Is the definition based on subsets (that both sets of subsets have to contain a particular subset --- in the case above it would only be $\{X,\emptyset\}$) or based on elements (I think then it should be: $X_1\cap X_2 =\{(x\in X\lor x\in A \lor x\in X\setminus A\lor x =\emptyset)\land\\ (x\in X\lor x\in B \lor x\in X\setminus B\lor x =\emptyset)\}$ )
The notation $X_1\cap X_2$ always refers to the intersection of the sets $X_1$ and $X_2$. That is, $$X_1\cap X_2=\{x:x\in X_1\wedge x\in X_2\}.$$ In your case, that would be $\{X,\emptyset\}$ (at least as long as $A\neq B$ and $A\neq X\setminus B$).
(There are situations where $X_1*X_2$ can refer to the set $\{x_1*x_2:x_1\in X_1,x_2\in X_2\}$ where $*$ is some binary operation. I've never see this usage when $*$ is intersection, though, and certainly no author would do so for intersection without commenting on what they are doing.)