My sophomore Latin teacher at Xavier HS always referred to Gaius,never Julius.
Your Latin teacher was a good teacher. Praenomen (Gaius) is what we would call our first name. To understand Roman names you must understand the Praenomen.
My favorite was the loser Commodus; Lucius Aelius Aurelius Commodus Augustus Herculeus Romanus Exsuperatorius Amazonius Invictus Felix Pius