If you add "using old-fashioned predicate logic" or "using classical logic" to the question, I agree with you. Vacuous truth is generally useful, but it's still an arbitrary choice. It's not a thing you can test like gravity.

This is why the problem statement bothers me. If you're going to contrive a puzzle out of pure logic, you had better constrain the world (ie, what logic system the "liar" uses). It's like formulating a geometry problem (behavior of parallel lines, sum of angles of a triangle, etc) and just assuming Euclidian space.