I begin to understand. We are divided by a common language. When a systems engineer speaks of saturation in an amplifier or material he means that the incremental gain decreases as the input signal increases, there is no implication of any physical mechanism analagous to a sponge taking up water. Saturation can be weak, strong or total. The logarithmic input/output characteristic is saturating because the incremental gain decreases monotonically with signal level. Over a wide range of sciences saturation is taken to mean the deviation from linearity, not just the flat bit (try googling, for instance, “onset of saturation”). It is important in the situation under discussion because runaway instability is impossible with a logarithmic characteristic in the feedback loop (unless of course there is a countervailing exponential characteristic).
As it happens, there is some relevance of the sponge analogy in this case, but it is modified by the competing process of spontaneous emission.