The only correction that needs to be applied here is this:

The photon is not absorbed by the electron, however, some of the energy that the photon possesses can be transferred to the electron.

The mistake the layman makes here is to assume that the photon is energy when in fact, the photon only carries varying amounts of energy. The photon is a wave/particle, it is not energy.

I'm afraid this doesn't square with Einstein's

E=mc² paper where Einstein said

*"If a body gives off the energy L in the form of radiation, its mass diminishes by L/c²"*. The photon is a wave where E=hf and E= hc/λ. It's a form of energy. If you take all the energy out of a wave it isn't there any more. Think about what would happen if you did another Compton scatter on the residual photon, and another and another. In the limit all the photon energy has been converted into the kinetic energy of electrons, and the photon is gone.