Looking for an approach to mathematical notation wherein the universe is divided into disjoint worlds.

Solution 1:

All three questions are purely about the conventions of mathematical notation; there is no need for creative mathematical metaphysics to resolve them. The first two examples you give show that it is not generally possible to extend notation in any way that one might find convenient, without bumping into already established conflicting interpretations. The solution is usually just to modify the notation so that confusion does not arise. Alternatively (and questionably) the old notation can be used anyway, and the reader specifically warned that it is not to be understood by its prior undesired interpretation. The third example points out an established abuse of notation that is already problematic. However, the abuse is so entrenched that anyone who introduced a new notational device to avoid the conflict would be regarded as pedantic, unless the mathematical context was one where both the set $X$ and its elements were in the domain of the function.