Are you suggesting that the men were known as Julius Caesar and Augustus Caesar? Because that's wrong[0]. Augustus used this name and it became a title for roman Emperors, in reference to Julius Caesar. If people are talking about Caesar, they mean Julius Caesar.

[0] https://en.wikipedia.org/wiki/Augustus#Name

I understand that people often refer to Julius Caesar as "Caesar", but to me at least, it mainly refers to the "title" (though that remains a surname, as succession was hereditary, mostly through adoption).

History is not my field, I am probably making multiple mistakes and oversimplifying, but I don't see something in that link that disproves what I said. Today, we often refer to Augustus as Augustus Caesar or Caesar Augustus (less frequently, Emperor Augustus). I also think it's interesting to point out Julius's first name in light of the previous discussion, as that's where the month name comes from.

If we're splitting hair, Octavian was adopted into the gens Julia, so he was a Julius as much as he was a Caesar, and his full name before ascending to emperor was Gaius Julius Caesar Octavianus.