Again, I understand your position and your logic, but the Romans themselves mark the Empire as beginning with Augustus.
History is self correcting. I can see that they thought of your definition. Why call something an empire if you don;t want to have an emperor. By your definition they could have taken over China and still wouldn’t have been an empire.
After you cited all those scholars it was very easy to come up with an alternative explanation. And of course I cited different definitions.
Was the British Empire ruled by a king or by parliament and a prime minister? Did it have an emperor?
Is the science settled? LOL.
Why would the romans mark the start with Julius when the senators killed him for it? They didn’t mark the start, Augustus self-declared, as Julius was doing. After Brutus and Cassius were defeated who would disagree with Augustus?
In my readings of Livy etc. The Romans took over various tribes, cities and territories and greatly expanded, as I referenced. They didn’t say hey look at the Roman Empire. They just called it Rome and referred to various provinces.
When we look back at the geography it is a simple mater to call it an empire whether the senate or the emperor ruled at a given time.