The name "Octavian"

[Gaius_Calvus]

This is how I understand the issue:

"Octavian" was G. Julius Caesar's (dictator perpetuus) grand nephew through his elder sister, Julia. "Octavian's" birth-father was Gaius Octavianus, which was also his (Octavian's) given name.

When Caesar was assassinated in 44 BC, he was childless (his daughter Julia, wife of Pompey, having died around 53-52 BC); she was his daughter by his first wife, Cornelia Cinna, to whom he was married at a very early age (at the same time Caesar was made Flamen Dialis by Gaius Marius). Cornelia Cinna died very young, and Caesar had no children by his second wife, Poppeia, (Sulla's grand-daughter).

Caesar therefore adopted Gaius Octavianus in his will, a not uncommon practice for a childless Roman nobleman (Caesar's family was patrician as well as noble). When men were adopted in this fashion, the traditional usage was to change one's name (Gaius Octavianus was now legally Gaius Julius Caesar) by adding one's former last name as a (first or second) cognomen -- thus, upon his adoption, Gaius Octavianus became Gaius Julius Caesar Octavianus -- and would therefore refer to himself (and be addressed as) Gaius Julius, praenomen and nomen, in polite conversation, or as "Caesar" if one were less formal (it becoming the practice at that time to refer to men by their cognomen only, as opposed to praenomen and nomen together, which was more correct/polite).

"Octavian" is the anglicized form of his second cognomen, Octavianus.

Thus, while "Octavian" is an invention of sorts (it is a diminution of Octavianus, his second cognomen), it (Octavianus) was truly a part of his legal name from the time of his adoption in Caesar's will until his death.