In order to elucidate the reasoning behind the differential equation yielding the "grey atmosphere" radiative lapse rate I now take the liberty to present some extracts from Ray Pierrehumbert's first online distribution of the book "Principles of Planetary Climate" dating November 11, 2009. In my view, this is the only understandable derivation that I have found. Take special notice of the formulas (4.6), (4.7), (4.8) and (4.29). Try to reproduce these with your own hand. I believe I have verified that, assuming that the net radiative flux q_T = (I+) - (I-) is independent of altitude, formula (4.29) yields the differential equation appearing first in the Ramanthan/Coakley review paper, except for some numerical factor which probably comes from some angular integral.