I'll go purist, and say that it's only a game if it has rules, objectives, strategic or tactical decisions, and multiple teams or players which all must follow those same rules, and attempt to reach the same objectives. If there is only one 'player' (A.I. can be players) then it is not a game, it is simply interactive text (using "text" in the broad literary sense). By this definition, Mario Kart is a game, but Super Mario Bros. is not because in the former, Bowser plays by the same rules as all other players, and in the later, he does not.
However, I think it's a bit pedantic, and we can use the word two ways, similar to the word "comics".
We use the word "comics" very loosely to define all manner of types of text, but technically, the term only really refers to books full of graphical jokes, of a sort which are very rarely published anymore. Most of what we typically call "comics" is actually more accurately called "Sequential art narrative", but this is really an academic term.
As a literature professor, I am careful to use terms like "Sequential art narrative" and "Interactive narrative" instead of "comic" or "game" when we study these works because it's an academic setting, and one should strive to be accurate in the classroom (in fact, I make distinctions among "interactive art", "interactive verse", and "interactive narrative" - plus all of those with and without the word "digital" - in my classroom), but when I talk to my friends about what I've been filling my leisure time with on the weekends, I talk about "comics" and "games" because that's the expected language, and using the more accurate language would be pedantic in that setting.
Is it important to know the technical difference among all these terms? I think so. Should we be attacking or even correcting others when they use them incorrectly? Probably not outside of a classroom or other discussion in which the distinction is important to the topic at hand.