I know that the interactions vary with the incident photon energy (from photoelectric effect to Compton effect to pair production).
What you need to understand is that the probabilities of the interactions vary with the incident photon energy. As the energy of the incident photon, $E$, increases, the probability of a photoelectric effect decreases and Compton scattering increases. PE dominates below 500 keV and depends on the material according to $Z^5$, where $Z$ is the atomic number.
Between 500 keV and 1100 keV, Compton scattering dominates the interaction probability with a material dependence of $Z$. There are a few other effects such as Raleigh scattering, electron resonance and Thomson scattering, but these are fairly minor in the the overall detector process.
As the excellent answers by Michael Seifert and Jared Popowski state, a single Compton event will always deposit less energy in the detector than the incident photon has. So, while PE does dominate at low energies, it is possible for a Compton event to occur, and this will produce a small Compton-edge in the spectrum even for low energy photons. Photons with $E>500$ keV are expected to have substantial Compton-edges, again associated with a full-energy peak, but lower than $E$.
Occasionally, the secondary photon will interact in the crystal by a subsequent photoelectric event within the time resolution of the detector system and the full energy $E$ might be "counted," but this isn't certain. So for mid-range gamma-energy photons, the Compton events will always show up at a lower energy than the associated full-energy peak.
OTOH, the probability of a large Compton edge increases with $E$, so the Peak:Compton ratio usually decreases with increasing $E$, as illustrated in the graphic you posted.
When $E>2m_e c^2$, the probability of pair production increases and probability of Compton scattering will decrease. Pair production has a broad maximum between 10-30 MeV. And as pointed out by the other answers, a single escape or double-escape maximum could occur due to the escape of the annihilation photons which follow pair-production.