Discrete Time Fourier Transform Pair Discrepancy
This is just normalized frequency versus un-normalized frequency. 

Check the capitalization of omega. The capital omega $\Omega$, usually refers to the normalized frequency (with $-\pi$ and $\pi$ corresponding to $-f_s/2$ and $f_s/2$) and the normal omega $\omega$, to the regular angular frequency.

Your two sources just use different convention. There is no contradiction.