John Fairbairn wrote:'proper' has just too many rather different meanings and nuances, none of sit well with go.
In English go terminology, the phrase "proper move" is used in the same function as "honte" used in English go terminology. Therefore, there are not too many rather different meanings and nuances. Too many rather different meanings and nuances can occur when "proper" is used as a non-go-term word (in go texts); this is not what I suggest. I suggest to continue using the full phrase "proper move" (or its grammatical derivates, such as "the move is proper") or "honte" by those preferring more Japanese words in English go terminology.
Its use stems from Kenkyusha,
I do not care, because, for English go terms use, I care for what has been English go terms use.
The bigger problem is that your definition would fit too many other types of play,
My definition is not broken down to axioms yet, so you are right that there are still other types of plays, and maybe too many of them. OTOH, since your description is significantly more ambiguous, this kind of problem is bigger for yours.
My definition relies on definitions of aji and "thick shape":
"_Aji_ lies in the latent, bad possibilities in a player's imperfect shape that the opponent might exploit to his advantage at a suitable moment." [10]
"A move creates _thick shape_ if it leaves behind little or no aji. The opponent cannot capture, cut or play painful forcing moves against it." [8] (A couple of the more fundamental terms are also defined.)
e.g. boundary plays.
My definition of proper move applies only to such boundary plays that "postpone the necessity for yet another local move until much later by eliminating aji and creating thick shape.". For this kind of boundary plays, it is right that my definition applies. It is also right that my definition does not apply to other kinds of boundary plays. So what is the problem?!
Worst of all, it would fit slack moves.
My definition of proper move would apply only to such slack moves that "postpone the necessity for yet another local move until much later by eliminating aji and creating thick shape.". IOW, you want to call moves 'slack' that a) postpone the necessity for yet another local move until much later, b) eliminate aji and c) create thick shape. Let us study your suggestion. (a) is not a slack aspect of a move at all; therefore, you are wrong calling moves given by my definition 'slack'. (b) can be slack if aji is eliminated prematurely; however, (b) is not applied alone, but (b) is applied together with (a) and (c). (c) speaks of CREATING thick shape, which is not a slack aspect of a move at all; therefore again, you are wrong calling moves given by my definition 'slack'.
In summary, my definition does not fit slack moves.
You need to read my definition more carefully, if you want to find part of the remaining gaps towards an axiomatic definition.