I'm not sure I agree with how you broke down art. There are many motivations for art, to which you mentioned a few, but those are not what makes art into art.
Art is the manifestation of our brain's capabilities when applied to a medium.
For example, painting is the use of pattern recognition on various objects to simplify them and recreate them through the mixing of pigments on a canvas. Drawing is the same, only using the medium of graphite, or charcoal, or pen, etc. Sculpting is likewise trying to simplify the patterns of the world.
The problem is when what we are using our brain to recreate is not something easily grasped. For example, what if you tried to paint loneliness? What if you tried to create the melancholy felt by the entire human population of the world as it faces it's possible impending doom due to its own actions? What if you just let your mind run wild and paint whatever it wanted to?
Similar processes to these lead to very abstract art that perhaps cannot be easily understood. Especially when it's using a medium that perhaps is not very well recognized, and departs a bit too far from the norm for most people's comfort.
It's still art though. I think, perhaps, a lot more is art than we are willing to accept. If someone makes a movie for example, it might not always be high art, but it is often art, even when we don't want to accept it. It's the director's vision applied to the world.
I don't think the motivation matters. Someone can create a beautiful piece of art for art's sake, or for a commission. In 1000 years, will we even know? Or will it be just another piece of art in a museam that people wonder the story behind?