First, your argument insists that if there is only one solution to the game, it is a puzzle. The trouble being, you switch your definition of "solution" to fit your conclusions rather than either adjusting the definition to fit your data or your conclusion to fit your definition. More specifically, having a linear narrative in a game that presents you with the same series of challenges each time you play is not necessarily the same as having a single solution. The solution to a challenge in Half-Life can vary significantly--who you shoot first, what boxes you hide behind, which weapons you use, your exact path through the map, and so forth. Viewing all of these as the same solution simply because the background mechanics are the same and the challenge setup is invariant every time you play the game leaves us with a definition of games that ignores much of the history of gaming--at the table and in front of a screen.
Second your argument insists that puzzles and games must be fundamentally exclusive and different. This is inconsistent with both popular and some analytical terminology. Terms can overlap; there's no rule that a puzzle cannot also be a game or that puzzle and game cannot be ambiguous, coexisting definitions. Some fields of study create such rules to assist in analysis, but especially since we aren't part of some rigorous body of games research we don't need to bother with such tidying up. Games can be puzzles or contain puzzles. Games can be stories, or contain stories. And so forth. This exclusivity is utterly unnecessary and completely out-of-keeping with general parlance about games within industry, within critique, and within the gaming community.