Let a finite group $G$ acts on a complex vector space $V$ and let $\mathbb{C}[V]^G$ be corresponding algebra of polynomial invariants. Let $f_1,f_2,\ldots,f_m$ be a generating set of this algebra of invariants. Consider the map $$ \phi:V\to \mathbb{C}^m, \phi(v)=(f_1(v),f_2(v),\ldots,f_m(v)). $$ Clearly, $\phi$ is constant on $G$-orbits.
Now, suppose that we have an another set of polynomials $g_1,g_2,\ldots,g_n$ $g_i \not =0 $ such that the map $$ \psi:V\to \mathbb{C}^m, \psi(v)=(g_1(v),g_2(v),\ldots,g_m(v)), $$ is also constant on $G$-orbits.
Question. Is the set $g_1,g_2,\ldots,g_n$ a generating set of the algebra $\mathbb{C}[V]^G$?
No, I don't think this is true. Consider the map $\psi(v)=(f_1(v)^2,\dots,f_m(v)^2)$. It is also constant on the orbits, since $f_i$ are constant. Polynomials $f_i^2$ do not generate the algebra of invariants. Actually, you can pick any $m$ polynomials from $\mathbb{C}[V]^G$ and assemble them into a map $\psi$ like above. It will be constant on orbits. but of course not every set of $m$ invariant oplynomials generated the algebra of invariants.