I do think there is a difference, but they are aspects of the same thing. One significant difference is that in the classical limit (ie taking the limit hbar

I do think there is a difference, but they are aspects of the same thing. One significant difference is that in the classical limit (ie taking the limit hbar -> 0) non-linear Compton scattering doesn’t produce energy loss for the electrons. The classical limit of radiation reaction (ie n photons scattering of an electrons m times) does produce energy loss (because the loss per scatter scales like hbar but the number of scatters scales like 1/hbar). The sentence you quote is about the energy of the photons produced (which determined by Compton scattering) but the energy loss of the electrons is what we call radiation reaction.