Perhaps he did lie about his age. Short of us being able to witness his birth for ourselves, we have to either take someone's word for it or collect enough information from sources we believe to be reliable to make our own calculation. This unfortunately is always the dilemma we face as Genealogists.
You can't reverse the endorsement of John's service in the Revolutionary War found in the respected sources you identified. What's more, you have only the slightest chance of finding any authentic information to contradict them.
I would identify him as a Revolutionary War veteran born about 1761. IE, I would be more certain of his service than his age.