Of course, if you have a formula describing the system's response to an arbitrary input $x[n]$, you will obtain the response to an impulse if you choose $x[n]=\delta[n]$.
However, I suppose that the idea of the exercise is to obtain a closed-form expression for the impulse response. You should recognize that the given input-output relation is just a convolution sum, and from that you can just write down the impulse response in a very simple form, without using an infinite sum.
You can also find that simple expression from your second formula by realizing that $\delta[k]$ is only non-zero for exactly one value of $k$. Depending on the value of $n$, that value may or may not be reached within the given summation limits.