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.