Byzantium is Roman Empire. The term “byzantium” is coined by german historian in 16th century to delegitimize late Roman Empire because they were orthodox plus they wanted to legitimize “Holy Roman Empire” as the successors of Rome which they could never succeed.
Western and Eastern Roman Empires were never totally independent states. Roman Empire just got divided into 2 administrative divisions to simplify micromanagement since it was not easy to rule such vast lands from one capital.
Western half of Roman Empire got invaded easily, again there was no “Eastern Roman Empire” but simply Roman Empire lost its western half.
Holy Roman Empire is a joke by itself. It is actually “Confederate Kingdom of Pretty German Princely States” who wanted to claim legacy to Roman Empire but they could not. First of all, east-west schism, papal supremacy is non-roman, pope was simply Bishop of Rome. Pope crowning someone as Roman Emperor is ridiculous. Also even coronation is barbaric by means of Roman Empire. Why ?
1- Coronation is given power from a higher source. Roman Emperors would ascend to throne, not get coronated which would seem as an insult.
2- Roman Empire did not have a “crown”. Ornate headgears were deplorable, they were simply random hats of emperors, power was by emperor, not the crown. Culture of coronation comes from Roman Emperor sending crown to his vassals, it is a diplomatic insult that said that headgear hold the power that emperor could grant to anyone he wanted.
Again, Emperors of Holy Roman Empire were not called Emperor or Roman or Holy by anyone pretty much, hence famous Voltaire quote “neither Holy nor Roman nor an Empire”.
Summary:
Western Roman Empire and Byzantium (Eastern Roman Empire) were both Roman Empire, they were not independent states but two administrative divisions of the same empire, one of these divisions got destroyed early on, leaving the other one and only sovereign of Roman Empire.
Holy Roman Empire was Confederate Kingdom of German Princely States with elective monarchy where Habsburg dynasty of Austria bribed electors all the time.
