Hello Boris,
Thank you for the comment, and for inquiring clarification on the definition.
I can totally related to your confusion about the "non-game context" clause in the definition of gamification. That is the most vague part of the definition, and there is good reason for it.
The reason is because we don't really have a really good definition of "games."
What is your definition of a game? I often hear people say that a game is a form of play or the activity that we engage in where the players follows a set of rules defined by the game and the outcome of the game is determined by skill and chance. If you believe in this or any other definition of games, then a lot of things can be a game, such as school, business, work, or even life in general.
So when I say non-game context, the games I was referring to are games like Angry Bird, poker, chess, basketball, etc., something that we do for fun, as opposed to work, school, business, exercise, etc. (even though some people may find these activity fun and think of it as just another big game). So it is very hard to define what a game is, consequently, I hope you see the difficulty in defining what is not a game.
They key point to take a way is that a good game should have all the necessary narrative, game mechanics/dynamics already and shouldn’t need to be gamified. If you have invented a kind of work that is so fun that everyone love to do, then you don’t need to gamify that work, even though it is work. But for that matter, those kind of work might as well be perceived as a game, and many people doing those work are probably thinking that they are playing a fun game.
So non-game context in my definition is rather vague, but what I want to say is that they are not already a game. At least not being thought of as a game "in the traditional sense (i.e. Angry Bird, basketball, poker, chess, etc.)" by most people even though anything we do can be thought of as a game.
Alright, I hope I clarify the confusion. Thank you for asking such great question.
I hope to see you around lithosphere more in the future.