It is not true that $\text{rank}_+^*(V) \leq \text{rank}_+(V) $.  In fact, an equivalent definition of the non-negative rank of $V$ is the minimum number of non-negative vectors (not necessarily columns of $V$) such that every column of $V$ is a conic combination of these vectors.  Therefore, the opposite inequality  $\text{rank}_+^*(V) \geq \text{rank}_+(V) $ holds.