I don't understand the reasoning of the author of this article, on page 26. How can he conclude that full earth-shine on the moon is 100 times brighter than full moon-shine on earth?
The author says that the difference between the factor of 100 and the geometric increase due to area and albedo is because of "atmospheric absorption and directional effects". The former is probably only about 20% or so (i.e. about 20% of moonlight is absorbed/scattered in the Earth's atmosphere). The latter effect could be quite big.
The reflectance of solar system objects shows an opposition surge at small phase angles.
It is difficult to see the full Moon at really small phase angles ($<2$ degrees) from Earth because you would get a lunar eclipse. However, it has been measured by spacecraft and according to Buratti et al. 1996, the brightness of the Moon increases by "more than 40%" between phase angles of just 4 degrees and zero.
When the Sun illuminates the Earth as seen from the Moon then it would be much easier to see at very small phase angles because the Moon's shadow is that much smaller.
I suspect this effect plus the ~20% absorption in the Earth's atmosphere is what the author of your reference is talking about.