Alexandra, you are of course technically correct, and I simplified things for the purpose of the quiz, perhaps too much. Although, I discovered when researching material for my article
Dalai Lama Succession Issues that there really is no separation between religion and government within the Tibetan system, so depending on how you define 'religious', the Dalai Lama is essentially the top religious and governmental leader. He doesn't hold any sway over the internal practices of each school, or their selection of lamas, but they all acknowledge him as the official representative of Tibetan Buddhism in general to the outside world, and he leads all multi-school religious gatherings. This current Dalai Lama has shifted that quite a bit, partly because of the Tibetan exile situation, so it's hard to say if that will hold going forward or not...