Let me please add to this thread on the meaning of the adiabatic coupling, the following preprint:
Geometry and non-adiabatic response in quantum and classical systems by Michael Kolodrubetz, Dries Sels, Pankaj Mehta & Anatoli Polkovnikov.
https://arxiv.org/pdf/1602.01062.pdf
I find it relevant, because all lectures & examples in this document are centre on the adiabaticity concept. (I am not a specialist in this subject, but I can follow well some examples)