The name "Octavian"

[Jona Lendering]

So "Octavius" name was simply Gaius Julius Caesar after the adoption then? Nothing to indicate where he came from, no second cognomens?

Exactly. He was not going to broadcast that he was not the real Julius Caesar. The ambiguity was his power.

I haven't done an exhaustive search, but I can't think of a single source that shows that these adoptive-cognomina ever existed except long after the fact... Jona, have you?

I think you are correct that most names were invented later, by people who had to distinguish people.