I would ask them to propose one that they think is satisfactory. Interviews shouldn't be game of secret answers that one is out for hunt. There shouldn't be any adversarial component in interviews. I often imagine candidate as 1-person startup who discovered this problem, perhaps not in very polished well defined form. As an interviewers I'm just thin air in his room observing the process of polishing, thinking and solving :).
I agree that "read my mind" is a terrible form of interview. One issue with the "let the candidate guide the conversation" approach as you are outlining here is that it is very very time consuming to train new people to administer this kind of evaluation in a repeatable, fair way. Giving the full definition helps the evaluation move along a bit faster, too. Then, it does not create an opportunity for evaluating "how do they explore the problem?"
