My impression having read this is that it is stretching credulity to breaking point.
There are no Quakers pre-1650 so there could be no Quaker records of John, except perhaps his death.
Any American one family genealogy from 1890 to 1910 immediately makes me nervous especially when it jumps back to England. That was peak fantasy genealogy period.
What we have here is a Quaker in Pennsylvania called Joseph Baker (at least he wasn't John Smith). You would need a lot of proof to get him back to his family in England. Everything else is late Victorian make-believe until proven otherwise.