The standard notion security for MACs is usually expressed by means of experiments like the following one (given an attacker A):

1. $\kappa = \leftarrow \mathsf{KG}(1^\lambda)$
2. Invoke $A^{\mathsf{TAG}_k(\cdot),\mathsf{VRFY}_k(\cdot,\cdot)}$
3. Output $1$ if $A$ queried $(m^*,\tau^*)$ to $\mathsf{VRFY}_k(\cdot,\cdot)$ s.t. $\mathsf{VRFY}_k(m^*,\tau^*) = \mathsf{Accept}$ and $A$ didn't receive $\tau^*$ by querying $m^*$ to $\mathsf{TAG}_k(\cdot)$

Now consider the following variation to point 3: Output $1$ if $A$ gives $k'$ s.t. $k=k'$

How could I prove that a MAC unforgeable according to the first game is unforgeable according to the second game too? Is there a specific $\mathsf{MAC}(\mathsf{KG},\mathsf{TAG},\mathsf{VRFY}$) that is unforgeable according to the second game, but not according to the first one?