Probably. You could also argue that if a government (state, local or federal) organization came to him and asked for a design, they're being "paid by him" since they're operating off of tax dollars.

That wasn't my point. The point is that in the real world, every designer (and developer) has a client, regardless of whether they're a freelancer, employee of a commercial company, or even a government employee. Who pays whom is unimportant in this context; what is important is that an agreement has taken place: the designer is to create something at the specific request of a consumer.

The clients have a need, and the designer's job is to meet that need. In the anecdote from the article, the professor is the client, and that client said "redesign a popsicle".

My point is that Kim, as talented as he may be, did not address his client's need in this case.