There are two major false premises that undergird your question.

Firstly, the Roman Empire never renamed itself ‘the Byzantine Empire’. They continued to call themselves the Roman Empire right through until the fall of Constantinople. The label ‘Byzantine Empire’ is one given by western Europeans, and utilised by historians, because it represents the fact that very significant changes did take place through Late Antiquity and the transition into the Middle Ages and Byzantine period. However, no – the Romans didn’t say ‘hey, let’s rebrand ourselves as the Byzantine Empire and relocate.’

Secondly, the Romans did not exactly ‘relocate’ and leave Italy for the East. They conquered the East, and incorporated the eastern half of the Mediterranean into their empire. That half of the empire was generally better developed, and better populated. It came to be of increasing importance over time. Meanwhile, for several reasons including food supply, population pressures, military incursions, Rome came to decline in prominence. If you think of the population and importance of cities as functioning like gravitational forces, the ‘centre of gravity’ moved east, and this was reflected not only in the building of Constantinople, but in the fact that Constantinople ‘succeeded’ as a city, and that several other important Imperial centres appeared in the east as well.

To go back to your analogy, it’s a little more like if the USA annexed Canada and Mexico, and all Mexicans became citizens of this ‘greater United States’, and then the centre of power shifted to Mexico City, and then 300 years later they lost control of half of the continental USA to roaming sea-pirates, but they kept calling themselves ‘USA’ and ‘Americans’, but everyone else started calling them something else.