A game has set rules and a skill or mindset to learn in order to play effectively.
A video game is a rather tenuous defintion. There are a myriad of genres but most do have a skill, whether intentional or otherwise, for the player to learn. Then there are such things as visual novels and virtual simulators which have no set skillset. I always think of the former being a step removed from quizzes and pick your own adventure novels and the later being a playground where one makes their rules of play if anything. Maybe interactive entertainment is more apt term for modern video games.
That said, I personally prefer games with rules and tight, responsive gameplay, defined rules and a very apparent skill curve to master but I suppose that doesn't make a game, a game in the video sense. I don't think of multi-player as mindless, if anything single-player is becoming more mindless as it's being a case of "Do what we want and we'll treat you with this." rather than anything with rewards exploration, experimentation or overall deviation. Multi-player, on the other hand, is a very much different people and different personalities and different philosophies clashing, often in real time, to acheive a shared or conflicting goal. I can't see that as mindless no matter the game.
Pixel Art can be fun.
