I can't speak for the question in your title, but I will try to answer this:

Why although the Eastern Empire Citizens was called Romans as citizens of the Roman Empire, today they are known as Byzantines, from the small village that was there with the name Byzantion, before Konstantinos the Emperor deside to change the capital city of the empire to that place and name it Konstantinoupolis?

The way it was seen at the time by the Romans themselves wasn't that there were two empires, a Western and an Eastern one, but that there was one Roman Empire, administrated by one emperor in the west and one in the east. Therefore, the fall of Rome and the western part of the empire was a blow to the emperor in the east, but it wasn't seen as an actual empire reaching its end. In fact, even Rome wasn't anywhere near as important as before; for decades, if not centuries before, the western emperor could often be found in other cities in the empire, such as Ravenna. So, life in the east went on, and though the Roman Empire they knew had lost a large chunk of land, and Rome still had symbolic value, they continued to see themselves as an empire that had been there all along. Only long after its fall, the view of a Western and an Eastern Empire came into play, and people saw the need to give the Eastern Empire a different name; the Byzantine Empire. Referring to themselves as Byzantines did happen occasionally, though mostly for e.g. poetic purposes, and not as a general term in daily use.

I feel like an answer to your first question cannot avoid a little editorializing but I'll try to shed light on it anyway.

First, there may be limitations to the premise. If you are from anywhere other than the footprint of Eastern Orthodox Christendom, it is likely that the historical narrative you've been chiefly exposed to (at least in regards to European history) is one that is Western European. Obviously, in Western Europe, their own history is most important to them, and that includes the legacy of Rome in the West. In addition, due to the colonial and imperial world dominance of Western European powers over the last few hundred years, the Western European narrative of history has spread much wider. However if you are from a place like Greece, Turkey, or something like that, the way that they learn about the Legacy of the Romans may be different from your experience.

Having said that, it would make a lot of sense that in the Western European tradition, the history of the Western Roman Empire would be of more significance. History, as a cultural narrative, is how people in a society come to understand where their society comes from, and for cultures that are Western European or borrow from Western Europe, then tracing some aspects of your society back inevitably leads to Rome.

Also, Western Europe experienced periodic surges of classicism in art, political philosophy, and architecture throughout the post-medieval period, most prominently in the Renaissance and during the Enlightenment. Classicism sought to revive various values, forms, and ideas from pre-Christian antiquity, which excludes the East-West split of the Roman Empire.

Also the history of the Western Roman Empire is full of lessons that remain relevant for modern societies. Above all others, the decay of the Republic into the Pricipate is the original cautionary tale for democratic societies. In addition, and indeed also related to the preceding, there is the cautionary tale of Rome's transitions from traditions of civic virtue to personal ambition and greed, from stoic modesty and honesty to corruption and decadence, and from civic order and participation to bread-and-circus populism. To some extent, Roman writers often gave their Republican past more credit than they should have, but there is more than a shred of truth to the narrative of decay of Roman society. Some of the founding fathers of the US in particular were quite enamored with Roman republican traditions.

Thank you very much for the replies.