Jay Puryear has many errors in his family history. He shows one of his own ancestors as being twelve years old when he married and thirteen when his first child was born. His info on William and Mary Pitt Puryear came directly from Grace Hollis Puryear's book "Puryear Kin and Kin to Kin". Grace got most of her info on this family by talking with descendents. I have concluded that she did not find the family in the 1850 census because the name was spelled Purearyer. As you noted, in the 1860 census the boys name was spelled Poyear.
If Walter Randall Puryear had been born in 1850 he would have been too young to serve in the Civil War. We know that he did serve so he must have been born before 1850.