I would agree that the definition of game can and should include titles that are not about amusement or making us feel good. I disagree that this means it should include Dear Esther. A better example of something that is definitely a game and definitely not about having fun is Pathologic.
For me, a game must have a goal. It can either have a definitive ending (when you “beat” it) or the goal is to reach some kind of high score. Also, there must be some sort of obstacle to overcome on your way to the goal.
I can’t for the life of me figure out why we stopped using the term “toy”. It used to be applied to titles like RCT (“software toy” was often used in fact), and it’s a great monniker. The glory in a game like RCT was always the sand box mode; the campaigns were not precisely afterthought but neither were they the meat of the game.