Are harmonics always the peaks in the frequency response plot?
Q: Are harmonics always peaks in the spectral plot?
A: No.  They may not rise above noise
Q: Are peaks in the spectral plot always harmonics?
A: No.  Peaks in the spectral plot indicate tones, or at least narrowband signals
Q: Can peaks in the spectral plot be noise?
A: Yes.
What's missing from your question, is what makes harmonics harmonics -- that spectral plot shows peaks that are not all at multiples of 770Hz, and are thus not harmonics.  Those peaks would be more properly called overtones -- but I don't know if they're intended as such, or if the plot is in error somehow.