If two sets have no elements in common, the regions do not overlap.

You seem to have skipped over the context in the Wikipedia page. That statement is true for an Euler diagram, not a Venn diagram. The next paragraph actually says outright that this is not the case for Venn diagrams. Here is that paragraph;

A Venn diagram, in contrast, is a graphical representation of n sets in which the n loops divide the plane into 2^n zones such that for each way of selecting some of the n sets (possibly all or none), there is a zone for the elements that belong to all the selected sets and none of the others. For example, if the sets are A, B, and C, there should be a zone for the elements that are inside A and C and outside B (even if such elements do not exist).

Note the “(even if such elements do not exist)”.

You can find the Wikipedia page on Venn diagrams here, which further illustrates this point (I have linked to the section that starts off by detailing how all relationships between sets are shown in Venn diagrams, even if they are empty, which is in contrast to Euler diagrams)