Lets place our context on what a tabletop rpg is about -how it is supposed to work in contrast with other activities officially acclaimed as games.
I'm still not sure of the purpose of the question, the thing that makes this something other than a discussion of semantics for the sake of semantics. Be that as it may...
Your statement above has a very simple flaw - "supposed to work", and "officially acclaimed" depend on the existence of some authority that defines such things. This authority does not exist.
Take, for example, the game of poker - the card game with perhaps the largest known number of variants. At a particular casino table, it is "supposed to work" a very particular way. However, how the game works in my living room is gong to be a rather different beast, and I'm not "wrong" to play it differently than at the casino table.
I submit that outside of mathematical "game theory", statements that a thing is a game have nothing to do with being "officially a game", and have everything to do with communicating. I think the phrase was first applied to D&D for sake of marketing - having only three words that get across a vague idea of what's going on is powerful stuff in sales.
Is it complete, unambiguous, and definitive? Perhaps not. But then, neither is the term "tomato sauce".
I'll deal with some other bits after I've gone and done my dishes -work before play, and all that
