i think its like this
SOTSP=senju and uchiha=full rinnegan
senju son = the sages body
uchiha son = the eyes
its like when we seen the uchiha sons eyes they looked like his fathers but they looked like they were slowly degrading into something else because he didn't have the the body(the senju son)(which is probably the key factor in obtaining the rinnegan), and over time without the senjus support(the body) they eventually degraded into a sharingan(the uchiha clan),thats why if you don't have senju dna the next best dna to use is uzumaki dna because it can substitute for the senju dna perfectly(because they are blood related to them)