Oh boy, Roman nomenclature? You are a daring one to go down THIS rabbit hole! It is confusing, definitely!

Put in the shortest possible terms, 'Gaius' was the personal name, and 'Julius Caesar' were the family names of the dictator and conqueror.

Gaius Octavius was the great-nephew of Gaius Julius Caesar. However, when Caesar's will named him as Caesar's adoptive son and heir, Octavius took on the family names - Julius Caesar. Thus making him, also, Gaius Julius Caesar.

For this reason, he's sometimes referred to as 'Octavian' or 'Octavianus' ('formerly Octavius'). Later in life, after he had solidified his hold on power, he badgered the Senate into granting him the name/title of 'Augustus' - 'Exalted One' - the name by which he's generally known as Emperor.

Later Roman Emperors would adopt the name 'Caesar' to identify themselves, initially, as part of the (theoretical) imperial family - later, 'Caesar' had become so widely used amongst the imperial family as a prestige title that it eventually became an office itself. In both cases, one could very easily simply say 'Caesar' to mean 'the ruler' in most cases, and colloquial use of the family name as a title predated its official classification as an office.