Wow there is actually someone else out there that thinks the 9-tomoe is actually the perfect Sharingan? I literally thought I was the only one.
Most people's assumption is that the 9-tomoe is actually a rinnegan + sharingan but the truth is we have only seen the color of the Doujutsu in the anime, and guess what? ITS RED! So if the anime is actually right on this one, I really think the 9-tomoe is the true sharingan (just as you said) and the Rinnegan is completely independent.
Think about it like this, how could the Sage have the Rinnegan and the Juubi happen to have not only the Rinnegan, but a combination of it and this so called sharingan that exists nowhere else but inside the Juubi? What I'm saying is that I think SO6P is the original Rinnegan user and Juubi is the original Sharingan user, simple.