1. Dude, the manga page you showed clearly has zetsu saying that Nagato is "from" the Senju lineage. Meaning the Uzumaki came from the Senju lineage. Meaning that the senju is a lineage of something bigger (so6p) but the Uzumaki is from the senju, i.e. the Uzumaki is a lineage from the Senju. The Uzumaki can't be both the original blooline and a descendent of the so6p at the time. You are ignoring an obvious fact pointed out in the very page you posted.
2. Even IF the Uzumaki were the "original untainted bloodline inheritance" as you put, that would mean that since the Uzumaki are untainted, then they would all be born with the RINNEGAN! Yet, no Uzumaki has unlocked the Rinnegan on their own in the history of Naruto, let alone be born with it. Madara told us plain as day in this chapter that he transplanted his Rinnegan to Nagato.
I don't know why so many people up here seem to have bought into what you've said. But it's wrong I'm sorry. You are correct in stating that a lineage is like a branch from a tree, but being of descent means the same thing as being a lineage to something. So you can't call the Uzumaki the true bloodline and a descent of the bloodline at the same time. That's like saying the Uchiha are the lineage of the Uchiha, makes no sense. The Uzumaki came FROM the senju. Not before them. It's in a manga and you pointed it out so nothing you or your diagram can say that will change that.
This is how it goes:
*So6p
-Uchiha lineage
-Senju lineage
-Uzumaki lineage
The Uchiha and Senju are the only direct descendents/lineages of the so6p. They weren't just branches, they were his sons. They were named as different descendents only because the sage actually split his powers between them, so neither had the full bloodline with them, which is why you need both to reach the true bloodline. Senju+Uchiha does not equal Uzumaki. Case closed
*Edit* Oh yea and I forgot, you need senju and uchiha dna to UNLOCK the Rinnegan. You don't need them both to use it. Anyone with transplanted eyes can use the eyes powers as long as they have the chakra reserves to do so. Which is why Kakashi can use the sharingan even though he's clearly not a Uchiha.