Sasuke represents the eyes of the sage while he's JUUBI JIN.
Naruto represents body of sage while he's JUUBI JIN.
That makes more sense IMO.
The manga never depicted if Hagoromo possessed or activated the RinneSharingan, even though he was the Juubi's jinchuriki, which seems to be the prerequisite(having the chakra from all tailed beasts by default of being the Jin of the 10 tailed demon), and seeing as Sasuke didn't fulfill that criteria, he just has a Rinnegan with Tomoe in it. Hagoromo, Madara, Kaguya and even Sasuke called his eye the Rinnegan, no one called it the Rinnesharingan except you.
The Rinnesharingan has 9 tomoes, Sasuke's rinnegan has 6 Tomoes (which indicates that his eye is fully charged, which is why his tomoe count goes lower than 6 at times, due it to it being used up), so your earlier statement that Sasuke can switch from 6 to 9 tomoe is false because he's never shown to do that, and even stated that his eyes are fully powered when at 6 tomoe at one point.