The exact disciplines are doing themselves a terrible disservice by muddying up established terminology like this (and "algebraic integers" are far from the only such case).
Mathematicians pretty reliably say "algebraic integer" or "integer in [some specific class of numbers that has non-rational integers in it]" when they are talking about the broader notion, and if they're doing something where that broader notion is often relevant they will generally say something like "rational integer" when they mean the narrower notion. So in practice there is seldom any confusion.
And algebraic integers really _are_ like ordinary integers in important ways. Inventing a completely new term would not obviously be an improvement.
It's not like this sort of thing is unique to mathematics. Once upon a time a "language" was a thing human beings used to communicate with one another. Then along came "programming languages" which are not languages in that sense. And then things like "hypertext markup language" which isn't a language in the programming sense either.
(Arguably this is partly mathematicians' fault since I think they were the first to use "language" to refer to purely formal constructs. But I think the use of "language" in computing arose mostly by analogy to human languages.)
And it happens plenty outside "the exact disciplines". A republican is someone who favours a mode of government that doesn't have monarchs, but if you call someone a "Republican" in the US you mean something rather more specific and a few "Republicans" would actually quite like a system hard to distinguish from monarchy. A window is a transparent thing placed in a wall to let light in, but a window of opportunity is something quite different. A czar is the absolute ruler of Russia, but when someone says (rightly or wrongly) that Kamala Harris was "border czar" they don't mean that. A star is a gigantic ball of stuff undergoing nuclear fusion and producing unimaginable amounts of energy, but even the most impressive rock stars don't do that, and some people called "rock stars" have never played or sung a note of rock music in their lives.
Roughly, a fooish bar will frequently be something like a bar except fooish, so not actually a bar. (Algebraic integer, multivalued function, manifold with boundary, etc.) On the other hand, a nonfooish baz when baz is normally fooish often means a not necessarily fooish baz, so a particular one might be fooish but we can’t assume that. (Noncommutative ring, nonassociative algebra, the very field of noncommutative geometry, etc.)
So, what term do propose they use for this? “Hutyfreklop” or “gensym_167336871904” would probably be unique, but wouldn’t tell anything about the subject itself, “roots of polynomials with integer coefficients and a leading coefficient of one” would get cumbersome soon.