Just a follow-up question though; in the non-CM case, is the Serre-Tate parameter a principal unit of K or of the maximal unramified extension of K (not in K)?

Some works which cited the Serre-Tate notes (in the manner mentioned above) that I know of are: Katz' Serre-Tate Local Moduli, Katz-Mazur's Arithmetic Moduli of Elliptic Curves and Peter Norman's Lifting Abelian Varieties. It is very likely that these authors have had direct access to the notes. Anyway, I do agree with you. We can only guess. Maybe it really was the Lubin-Serre-Tate notes.

Please excuse my impertinence towards homological algebra. I hope the latest fix is acceptable enough.

Thanks for pointing that out and for the idea. I already fixed the definition.