I don't you can believe you can discuss the concept of planets "orbiting" stars vs. stars "orbiting" planets. I believe in classical Copernican equations the two bodies effectively orbit each other. You seem to be thinking along the lines that the orbits must be roughly circular. Comets obviously demonstrate that that doesn't have to be the case. But all of the planets with "moon" systems show that one generally thinks of the least massive objects orbiting the heaviest object. I suppose the only case in our solar system where that might be a little less clear is in the case of the Pluto & Charon where the masses are closer to each other. I think the "center" of the orbit of the bodies is determined by the distribution of the masses. For the sun and most of the planets I think the center of the orbit is effectively inside the heaviest object. However with Pluto & Charon that may not be the case (but I'm not sure).