Comment above posted as an answer.  No, that is not true.  One counterexample is when $X=\mathbb{A}^1 = \text{Spec} \ \mathbb{C}[x]$, $m$ equals $1$, $\phi(\lambda,x)$ equals $\lambda x$, and $\psi(\lambda,x)$ equals $\lambda(x-1)+1$.