What is a polynomial with integer coefficients called? What is the proper mathematical name for a polynomial with only integer coefficients?
Having done some searching myself, I found "numerical polynomial" but these have integer values for integer parameters, and don't necessarily have to have integer coefficients.
As a related question, is there a name for polynomials with only rational coefficients?
 A: It is common to use the phrase integer polynomial to mean a polynomial with integer coefficients, consistent with describing a polynomial with real coefficients as a real polynomial or a polynomial with complex coefficients as a complex polynomial.
Any ambiguity arises from the dual nature of polynomials as formal expressions and as functions.  As your Question notes, there can be rational polynomials which take only integer values at integer arguments, so the best practice is to introduce (define) your meaning for integer polynomials along with notation, e.g. an integer polynomial is an element of $\mathbb Z[x]$ as @J.W.Tanner suggests (in the univariate case).
Likewise we can write $\mathbb Q[x]$ to clarify the meaning of "rational polynomials" as polynomials with rational coefficients.  It is always permitted to an author to redefine terminology (and notation) to suit the immediate needs of exposition, but incumbent on them to provide a clear definition and point out the (potential) deviation from convention.
