Yes, that William is my great great grandfather. He first took his family to Wallsend in Northumberland in late 1890/1891, but were back in Aberdeen by Nov. 1892 when their son William was born. My grandfather, Hardy was born in Leith (Edinburgh) in 1897.
They went back to Wallsend around 1899 and stayed a couple of years before again returning to Aberdeen by the birth of Jemima in Jan. 1906. Finally they went to the US.
My father is still in touch with Arthur and Joan Becker.
PS If you have a photo of the 1862 William I would love to have a copy. I have photos of Williamina and all the rest of the family.