Is the Compton scattering elastic or inelastic? Because the kinetic energy is conserved (in addition to the total energy conservation) so it should be elastic, but the energy of the photon is changed, so from that point of view is inelastic. I have found both definitions online, which one is correct?
When you are considering scattering, you are looking to what happens to the incoming particle. In Compton scattering, the incoming particle, the photon, comes out after the scattering process with less energy than it started with, and so the scattering is called inelastic; whereas, with Thomson scattering, since the energy of the photon does not change, that would be termed as elastic scattering.
It might be that the confusion arises because of the term "elastic" alternatively being used for collisions when the total kinetic energy of the interacting particles is conserved?
It is often called an inelastic process but this would result in an excitation of inner degrees of freedom of the electron. These an electron does not provide, therefore it cannot be an inelastic process.