I do admit, we were def talking about two different things at first. But when we were on the same page, you must admit your whole arguement depended on the words "distant blood relative" to be exactly the same as its definition.
I just wanted to point out it is very unlikely for the simple fact that the way you explain it, the Uzumaki should be equally "distant blood relatives" with BOTH Senju AND Uchiha. But the Uchiha are NEVER referred to as "distant blood relatives" of the Uchiha! This fact alone makes your point invalid, think about it.
In other words, your trying to be way too technical for a manga that never mirrors the real world 100%. If ALL CURRENT UZUMAKI descended from the Sage's "clan members" and not specifically his Younger Sons lineage, the manga should be referring to BOTH Senju AND Uchiha as the Uzumakis "distant blood relative". But the manga has never refer to the Uchiha as the Uzumakis distant blood relative--but according to your logic, it should.
Like I said, if your correct and the Uzumaki Clan is the Sage's Clan (and not a clan branched off from the Senju Clan) please explain to me WHY the manga never calls the Uchiha distant blood relatives of the Uchiha clan. But you can't, can you...?