The good in "public good" must be in the public domain not the private, or alternatively of the public as a whole, the definition that the test says is correct fails to include that restriction.
I am not trying to defend the test, and certainly not attacking you, but "public goods" is a widely understood and accepted term in economics. The definition is not at all controversial, and the relationship between public goods and government funding is a pretty bright line between liberal and conservative outlooks. Because of the way it is presented in the quiz, I would be very surprised if I were to learn that the authors are not conservative, because the question and answer, while entirely objectively correct, also support a conservative view.
To suggest that "public goods" are defined by government funding stands a generations-old debate on its head.
Again, what is meant or said or achieved by this quiz is irrelevant, but I'd really recommend that you do a little more reading before you have this discussion with a liberal.