Orac I'm no relation, just interested. I do family trees as a hobby. I got the info from a copy of his birth cert. Also I got a copy of his parents marriage cert.
He may have been baptised on the fifth. It was sometimes considered to be a more important date.
I had noticed that a search on the internet would come up with that different date. It's not surprising. If you do a search for John Lennon's family tree, there's not one site that has it correct or even one book printed that's right. These mistakes get handed down.