Well, no. "Julius Caesar" with no other qualifier is Julius Caesar, even if other rulers also used the name. Otherwise it would be difficult to differentiate them.
Well yes, that's quite literally how it worked. When Octavian was adopted he literally became Gaius Julius Caesar and technically lost the "Octavianus" part of his former name; roman names did not work like our modern day names.
The article is well constructed for its intended audience without getting into largely irrelevant details about the story.
It's accurate and purposeful. It's certainly not incorrect by any measure.
That's how it worked then (for Romans), but it's not how it works now. So when the writer refers to Julius Caesar without any other qualifications, he refers to the adoptive father of Octavian. It's a minor nitpick though, hardly worth bringing up, but it is correct.
edit: as someone else observed, none of these were part of the late Roman Empire. Octavian and Nero lived in the early Roman Empire, and Caesar lived during the last years of the Republic.
