Interesting thread.
The Book of Common Prayer [Church of England] states that
"the Minister of every parish shall warn the people that without great cause and necessity they procure not their children to be baptized at home in their houses. But when need shall compel them so to do, then Baptism shall be administered on this fashion"'
It goes on to say
"And let them not doubt, but that the Child so baptized is lawfully and sufficiently baptized, and ought not to be baptized again. Yet nevertheless, if the Child which is after this sort baptized do afterward live, it is expedient that it be brought into the Church, to the intent that, if the Minister of the same Parish did himself baptize that Child, the Congregation may be certified of the true form of Baptism, by him privately before used. It then goes on to say how to publicly certify the original baptism.
Hopefully this explains the confusion over second baptisms.