For simplicity this is about polynomials in just two variables.

Any $f\in\mathbb Q[X,Y]$ can be written as a linear combination of monomials $X^iY^j$ and therefore as a sum of polynomials $p_{ij}\in\mathbb Q[X^iY^j]$ over one variable:

$$\displaystyle f(X,Y)=p_{00}+\sum_{\gcd(i,j)=1}p_{ij}(X^iY^j).$$

My question: is the subring of all integer valued polynomials $f$ over two variables identical with the subring of all sums of integer valued polynomials $p_{ij}$ over one variable as above?

An integer valued polynomial in one variable is a polynomial $p$ with rational coefficients such that $p(\mathbb Z)\subseteq \mathbb Z$. And corresponding for polynomials over several variables.

It might be some abuse of language to call $p_{ij}$ polynomials over one variable, they may rather be polynomials over one variable applied to monomials $X^iY^j$.