Never said they weren't. Read my post again, I said "fan databook translation," the official databook never came out yet.
Well, considering the databook, which is canon even if you want to deny it, calls it a RinneSharingan, you're wrong on this account. And Powers of Sharingan =/= Sharingan.
Again, a fan translation is not canon. Manga says sharingan, so your opinion means nothing against kishi's words.
Rennigan has no correlation with Sharingan. Rennigan is the product of Indra + Ashura Chakras. Meaning Naruto and Hashirama could obtain it too, with no Sharingan. Likewise, even if some fodder took Sasuke and Naruto's DNA, that fodder would awaken Rennigan due to now having Hagoromo's Chakra. Even Momoshiki has Rennigan with no Sharingan relation.
I'm going to break down this fanfiction one by one.
1. Hashirama and Naruto cannot awaken the rinnegan, that is garbage fanfiction. The rinnegan can be attained by hagormoo's chakra purely and only Indra's transmigrants can awaken the rinnegan like black zetsu said because they are the only ones with chakra in the eyes.
If naruto and hashirama gained indra's chakra, they will not gain the rinnegan. They don't have the ocular power of the sage. Saying hashirama and naruto can awaken the rinnegan is as stupid as saying Sasuke and Madara can awaken Rsm like naruto did.
Eyes are Indra and Ashura are body, don't get your conception altered like that.
2. You used Momoshiki as an example yet he attained the rinnegan without Indra+Ashura. So take that into consideration.
So obviously gaining Hagoromoo's chakra doesn't grant you the rinnegan otherwise Kakashi would've some how gained the rinnegan once he got the double mangekyo.