The 1958 death certificate for the younger William lists him (and his father) as William Newman Barnes. His mother is listed as Margaret Walker, and the birthdate is 20 Aug. 1871, so those fit. Not sure why they became Barnes?
The younger William's petition for naturalization also says William Newman Barnes with the same birthdate and place as the birth record from above. It says he came from Dunham, Canada to the US via railway on 15 Mar. 1901.