I've discovered recently in academia that I have an odd habit of splitting terms into sub-terms, usually into a pair. Analyzing this question, I noticed that I've done the same thing to my definition of "Game". With what I've compiled from the various videos I've watched and discussions I've had on RRU, what a game is to me can be split into two concepts: Games-as-play, and Games-as-narrative.
Games-as-play are completely recreational activities with an artificial failure state that the player must overcome to get their satisfaction from the game. A good* game-as-play presents each problem in a way the player can understand it, and provides the right level of challenge and freedom so that the player feels challenged and not frustrated.
Games-as-narrative are stories with interactive elements. They will not progress the story if the player fails, or give the player freedom to choose the story's outcome or events. A good* game-as-narrative will leave the player satisfied with the story, the fantasy or world the game establishes, the role they got to play, and/or what they learned from the story.
Most games are a hybrid of these concepts, though there are pure forms of either.
*Quality is subjective and satisfaction will vary between players.