Okay, I think maybe there's a bit of a consensus forming. Regarding senses (1,2,3,4) from the OP:

There may be a distinction to be made here, but it seems that such a distinction isn't really the sort of distinction which merits having a tag of its own.

As per Rodrigo de Azevedo's suggestion, this sense is basically a synonym for the applied-mathematics tag. Personally, I could imagine asking, say a question about applications of set theory to the real world. In such a case, I might think to add an "applications" tag while it might not occur to me to add an "applied-mathematics" tag (even though it really *should*). But if I understand the tag synonym system correctly, it serves exactly this purpose -- if I input "applications", it will be replaced with "applied-mathematics".

We should really have some tag guidance for the "applied-mathematics" tag too.

This seems to be just the union of (1,2).

To the extent that this is on-topic at all, it should be a different tag.

So following Rodrigo de Azevedo's suggestion, I'd propose that the thing to do is:

Add some tag guidance for "applied-mathematics".

Make "applications" a synonym of "applied-mathematics".

Whenever an old question comes up which is now tagged "applied-mathematics" because it was originally tagged "applications" in sense (1) or (4), remove the tag. But probably we don't have to go do a mass tag removal all at once.

`{applications}`

mean sense (1) makes it so broad that it applies to every question. What MO questiondoesn'tinvolve applying mathematics to mathematics? $\endgroup$ – LSpice Mar 10 at 14:43isa reasonably well-defined class of questions of the form "What are some applications of mathematical theory X to mathematical theory Y?" (where Y is not "contained" in X -- such a question may also quantify over such Y), which is substantially smaller than the class of all MO questions. So it was too sweeping of me to claim that there's no distinction to be made at all about OP's (1). Now the question is: does it make sense to have a tag which signifies that a question is of this class? I think not, but I could imagine being persuaded otherwise. $\endgroup$ – Tim Campion♦ Mar 10 at 17:41becomesa tool of "applied math"! $\endgroup$ – Tim Campion♦ Mar 10 at 17:50