That William Carter could be his father as is name was also William Carter (Coachman) the son is listed as a Groom on his wedding certificate in 1861.
Yes, not very original keep naming each subsequent son William!
William and Mary go on to have a son and name him William too (Aaaargh) B: 1862
This young man is listed in the 1871 census as living with Grandparents William Stafford.