What does the phase spectrum show?

The magnitude and phase spectra are calculated from the complex output Xf using abs(Xf) and angle(Xf), respectively (see Example 3.3). Again, the angle routine gives phase in radians so as to convert to the more commonly used degrees scale by 360/(2π).

The phase of a signal generally refers to the timing of the signal (or how two sinusoids line up) as you posted in your question. But you are asking about the phase of a signal in the frequency domain (i.e., after an FFT operation). The FFT function computes an N-point complex DFT.

By the common words, the phase spectrum shows the phase shifts between signals with different frequencies . The very simple example is the chromatic dispersion. Assume the signal has definite phase shifts at the input in some volume with dispersive medium.