I suspect that the connection of John and Thomas Carter of Amelia to a Jesse Walton also of Amelia involved two Walton men named Jesse (or Jessee).
The Jesse Walton connected to Thomas Carter was in fact his half brother, both sons of Jane Filppen Walton Carter.
The Jesse Walton connected to John Carter was most likely the Jessee Carter who appeared on the the First Census as residing in 1785 in Amelia as head of a household with 9 whites, 3 dwellings and 3 other buildings.
The Walton connected to John Carter was a surety on a Carter bond signed in 1812, an unlikely act of one who was charging his sister board after she moved to live with him following the death of their mother.
Does the 1803 Amelia deed from James Vaughan to John Carter indicate where Carter lived at the time?