Correct name of Prescott Bush Sr

For many years people have fallen into the belief that Prescott Bush’s first name was John and yet I have never seen proof of this. They also have said that his son John was a Jr and again no proof of this. From my research I believe that Prescott’s first name is actually William and his son, William Prescott is the Jr.
1. 1840 census lists Prescott Bush Sr in one household and Prescott Bush Jr in another household and then John Bush as a separate household in Stewart Co Ga.
2. . from the Greenville Advocate December 20, 1866-1868, Apr. 4, 1867 This day came W. P. Bush, Admn for the Estate of W.P. Bush, Deceased April 2 1867. This is Prescott Bush Jr who is deceased and his nephew William Prescott Bush (John Bush/Fannie Johnson) who was the administrator of his estate and named after him and his grandfather.
3. John Bush is referred to in his estate papers as Senior and his son John Bush/Lucinda Winzer is called Jr. at this time no middle names are given and is the only time I have seen John listed with a title. John/Lucinda had a son named John/Mollie Shanks also and is called Jr. This is my great grandfather and I have a picture of him with the name John Henry Bush Jr on it. I also have an uncle who is named after him and according to what his father told him (my grandfather) this is the name handed down through the generations.; but again no actual proof.
As researchers we constantly demand sources and documentation on everything for solid proof; but there has never been proof; just the assumption that Prescott’s first name was John. I have always been curious as to why.