I was always told that the difference between art and design is that the artist creates the problem, and the designers solve them.
I thought it followed the Socrates tradition in that the true philosopher is the one asking the questions, and it is the role of the student to answer them.
I wish I remembered who I am quoting here
I wish ancient Greek techne τέχνη hadn't gone through the split that left "art" on one side and "technology" (or work?) on the other.
The split of art vs. design you're talking about or one of the many ways to divide the act of creation into a classical/romantic divide or one of the many other ways to describe it should be considered harmful.
And I'm not trying to split hairs here but wishing the dichotomy you're talking about didn't exist and encouraging folks not to frame the world that way.