We introduce the `Height-moduli' which allows us to differentiate between integral points and rational points on proper algebraic stacks with respect to stacky height functions formulated by Ellenberg, Zureick-Brown, and Satriano. Focusing upon Weierstrass equations and their moduli counterparts of cyclotomic stacks, the distinction naturally accounts for the origin of second and third main terms in the sharp enumerations of elliptic curves over global function fields ordered by discriminant. This is joint work with Dori Bejleri (Harvard) and Matthew Satriano (Waterloo).