Power/Amplitude Response of a system
Your second plot is the square of the magnitude of the frequency response in dB, which indeed doesn't make much sense. You must use $|H(j\omega)|^2$, and not $(20\log_{10}|H(j\omega)|)^2$, as you did. Of course, $|H(j\omega)|^2$ will not give you any new information compared to $H(j\omega)$.