I'm delighted to see you get into this topic since it is something that I've been thinking about for the last 35 years since as an industrial design student at a fine arts school. Being there instead of engineering school was my first clue. And I always described the difference between design and art as intent BUT exactly the opposite of your view. As a designer I have the intent to fix a problem through an object (or service) by making it easier to understand and use. That's intentional. What I describe as art is something that flows out of you. Doesn't have an intended audience. It just needs to come out. It is not necessarily trying to fix something but it is a refined punch of the feeling that the artist is already feeling intensely and cannot refrain from putting into another medium maybe to have a little relief from the intensity of the feeling.
But to me, a designer, a feeling is a tool that I can use to communicate. You need the empathy to anticipate the impact it will have on the user.