Who was the first us president to adopt the informal version of his name?

Jimmy Carter is the best answer. He always signs as"Jimmy" or "J" and appeared on the ballot as Jimmy Carter.

John Kennedy sometimes signed his name as Jack, but on legal documents he signed his real name. Theodore Roosevelt is often referred to as Teddy, but I do not think he ever signed his name that way-- in fact, I read somewhere that he did not like to be called "Teddy".
Jimmy (James) Carter goes by his nickname. John Kennedy often signed his name as Jack.