When notion of polynomial and polynomial function coincide. Before I never thought about the difference between a polynomial function and polynomial over a certain field $K$. Given examples such as when $K = R$ or $K = C$, we see that sometimes the notion of polynomial and polynomial function coincide in a sense that if $f(x) = a_nx^n+...+a_0$ is a polynomial function, then $$ f\equiv 0 \iff a_n = a_{n-1} = ... = a_0 = 0.\tag{1} $$ Sometimes it does not coincide, as we can see when $K = Z_2$. My question is, what are fields for which the notion of polynomial and polynomial function like in $(1)$ coincide called, and what are their basic properties? I suspect that $(1)$ never holds for a finite field.

They coincide iff the field is infinite. Over any finite field whose elements are $r_1,\dots,r_n$, note that the polynomial $(x-r_1)(x-r_2)\dots(x-r_n)$ vanishes identically but its coefficients are not all $0$ since the leading coefficient is $1$. On the other hand, over any infinite field, if a polynomial of degree $n$ cannot have more than $n$ roots (since each root gives a linear factor) and so a polynomial with infinitely many roots can only be the zero polynomial.