I think you're conflating a word, a concept, and models of reality. Scientists generally have no trouble with any of these, or with keeping them appropriately distinct. For instance, the zero of potential energy is typically assigned based on convenience for calculation (and such is stated). Nobody's the least bit confused, except the inattentive student, I suppose.

The only example you actually gave--dates--has nothing whatsoever to do with scientists and instead is a concession to pre-scientific convention!

What you say about number theory isn't even correct. There are number theory theorems about positive integers, all integers, non-negative integers, rings Zp where p is prime (numbers 0 to p-1, inclusive), and so on; and anyway, basically all science is done using a model of the reals instead. What you express has basically no relationship to reality.