Second pre-image resistance vs Collision resistance
The difference is in the choice of $m_1$.


In the first case (second preimage resistance), the attacker is handed a fixed $m_1$ to which he has to find a different $m_2$ with equal hash. In particular, he can't choose $m_1$.
In the second case (collision resistance), the attacker can freely choose both messages $m_1$ and $m_2$, with the only requirement that they are different (and hash to the same value).


(From this, it is also obvious that collision resistance implies second preimage resistance: An attacker can just choose an arbitrary $m_1$ and compute a second preimage $m_2$ to obtain a collision.)