For example, I know for a fact that $\phi^3$ in $D=4$ can be made finite renormalizing only the mass $m^2$. Is it possible, by a suitable field redefinition, transfer this renormalization to the coupling?
If this is true, in the first case above the beta function would be zero (coupling doesn't change with scale), and, in the other, it would be non-zero. But that contradicts the result given in the answer for this post (given that beta function is independent of the parameter we choose to renormalize in a Lagrangian) Super-renormalizable theory and $\beta$-function, where we have a general result for the beta function of $\phi^3$ in any dimension. In particular, in $D=4$, the result shows that the beta function doesn't vanish.
That said, I think something is wrong with my reasoning and would like that someone pointed this out concerning the aspects that I mentioned above. More specifically, about the choice in what parameters to renormalize in a super-renormalizable theory and if the beta function should depend on what parameter we choose.