What is the purpose of "x" in the Venn-diagrams depicting categorical propositions?

Question

See the I- and O-diagrams of this article. The "x" does not make sense the way it is formulated. Every member of the intersection is a member of the intersection, obviously.

The "x" cannot be supposed to mean that the intersection is inhabited by members. If the intersection was empty, there would either be no intersection in the diagram, or the intersection would be greyed-out. The existence of the open intersection necessitates the existence of intersectional members.

I guess the "x" is supposed to bring attention to the subset referred to by the diagrammed proposition, and maybe its existence implies that the subsets not enclosing the "x" are potentially uninhabited. This was not explicated in the article as far as I know, and if that's the case, then a far clearer language would simply be to add a third colour.

Consider this: white, grey, black.

If the proposition asserts a (sub)set is empty, that (sub)set is black. If the proposition asserts a (sub)set is inhabited, it is white. If the proposition does not make any assertions about a (sub)set, it is grey. This language would thus produce these Venn diagrams for I and O, the proposition being "Some cars are fast" and "Some people are not rocks":

This language would produce the following diagrams for "All cats are animals" and "No fish are birds":

Not only does this remove the nonsensical "x", but it also removes the homography between white meaning the (sub)set contains all the nouns of the proposition, and white meaning the (sub)set is not addressed by the proposition.

Answer 1

The "X" means that the set is not empty. In other words, there exists at least one member of that set. The idea being, I imagine, that putting an "X" in that area indicates that something is in there.

For example, this is the diagram for "some A is B":

Which, of course, means "there exists some X that is both A and B", or "the intersection of A and B is not empty".

The existence of the open intersection necessitates the existence of intersectional members.

It does not. In general, the actual size of the intersection area in a Venn diagram doesn't really mean anything; what matters is what elements are placed in it. And that could very well be "nothing".

I guess the "x" is supposed to bring attention to the subset referred to by the diagrammed proposition, and maybe its existence implies that the subsets not enclosing the "x" are potentially uninhabited.

Correct.

Sure, you could also introduce a third color for this. But I think drawing something inside the set to indicate "this set is not empty"—or in other words, "there's something inside this set"—is clear enough, especially given the examples.