His name was changed to Peter (by Christ Himself) and He was the first pope named Peter. Hence: Pope Peter I.
By way of further illustration, since World War I was not known as such until during/after World War II, therefore we cannot call it World War I...at least by your logic.
Peter never signed any letters as the Pope of Rome. Your example of ww1 and ww2 is not valid, David called himself the King he was not named the King after his death as King David the 1st.
Now James the brother of Jesus did use that name and stated so himself that he was the brother of Jesus one of a number of brothers and sister.