Power Spectrum and Power Spectral Density
What is the link between them? Power should not be function of
  frequency (it is the integral on all frequencies of power spectral
  density)


As far as a Spectrum Analyzer is concerned, to display the power spectrum in dBm, what it does it to take the FFT and compute square of magnitude $P_n = 20  \times log_{10}(|X(f_n)|^2*1000/100)$ with $f_n$ being the discrete frequencies separated by resolution bandwidth (see it is set to 1kHz for this case). I multiplied by 1000 since unit is in dBm and I divided by 100 because to get to power, I have to pass voltage $X(f_n)$ through a resistance of $50\Omega$. ($\frac{V^2}{2R}$).

To derive power spectral density from power spectrum, I am sure it is just dividing power in each bin by RBW. $P_n/f_{rbw}$. Though I am not very sure whether it is the (dBm value divided by rbw) or (dBm value of linear power value divided by rbw).


  I think this representation has no sense: the ordinate should be the
  power spectral density, not its integral in a neighborhood around the
  frequency considered


Well, in this case it is the latter. I think that is how all commercial spectrum analyzers represent spectrum (if you choose units as dBm or Watts). The comment by MBaz correctly mentions this.

Note - The final plot will be slightly different because your FFT size output will be different from number of trace points used to display on the screen. In order to combine several FFT points into 1 trace point on screen, a detector logic is used.