Inverse DTFT of $e^{-j\omega}$
Your second try yielded a normalized sinc function. That function has a zero-crossing whenever $\pi(n-1)$ ia a nonzero multiple of $\pi$, which is for every $n\ne 1$ (assuming n is an integer) and is equal to 1 when $\pi(n-1)=0$.
Therefore, both solutions are the same.