Earlier quoted context omitted.
> In Bavaria where I live, the informal convention is also last name first It's probably a Roman thing. Julius Caesar was actually "Caesar, of the gens Julia", so last-name first. In Italy we also used to list last-name first, particularly in official and military communications. This has changed significantly in the last 40 years, particularly in everyday bureaucracy, but it's still used by police and other bodies.
Gaius (also written Caius) was his given name, Ivlivs (Julius) the family name, Caesar the cognomen. In inscriptions (short): CAIVS IVLIVS CAESAR In speaking he would (in English;)) rather formally introduce himself as "Gaius of the Iulii, called Caesar". His mother called him Gaius, his peers called him Caesar once he got or choose?) this name.
His father was 'Gaius Julius Caesar' his grandfather (supposedly the first 'Ceasar') was 'Sextus Julius Caesar'.
The Cognomen is presumable something that evolved as a way to tell apart people belonging to different branches of the same family/clan because the number of first names was very limited.