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 question doesn't involve applying mathematics to mathematics? $\endgroup$