Roman Empire
The Byzantine — Eastern Roman Empire — Greek — Eastern Administrative Department of the Roman Empire
— Empire's and its people name from 324 to 1453
— Officially known as Roman Empire (324-1453) it is the Empire that flourish from the ancient Greek city of Byzantine, later to be Constantinople (after Emperor Constantine moved the Capital, alternative name "New Rome" or "The City" or "Is tin Polin" —meaning To the City and which form the modern name Istanbul). Its civilization, Greek (and Greek-Latin later on), referred Byzantine from the name of the ancient Greek city already existed, to specify that period of time in Greek history. It is the absolute and only continuation of the Roman Empire. The Empire had two main administrative departments, the Eastern Roman Empire and the Western Roman Empire. On the 4th century the Western administrative department of the Empire decline and cease to exist. Later on 9th century a new Empire created with the name Holy Roman Empire, however that entity has nothing to do with the Roman Empire and its history. The Roman or Greek or Byzantine Empire existed for more than a millennia and it is the longest lived state in the world history.
The status of the Roman citizen initially reserved for Greeks and Italians (Latins) only in the first centuries. Greeks never called themselves as Byzantines or Eastern Romans or else during that time, Greeks knew they are Greeks and they called themselves Romans or better "Romios" (which later on evolve as a word Romiosini meaning the spiritual level also, this is another topic on each own), but it is a term (Byzantine) used as indicative only, and enter in literature only after the 15th century, in English and French literature was more common to refer as the Hellenic (Greek) Empire and in German literature as the Byzantine Empire.
After 1821 as the successful Greek revolution goes on against the Ottomans and after, Greeks liberate south Greece at the time and create their Institutions after 4 centuries of Ottoman Tyranny. Of course the nation have been in the dark for centuries with no related Greek education (which is truly amazing that survived intact and restored -or nearly- today in such short time and advanced - and even overpass the invaders status long ago). Greeks looked into the multiple terms as they have been called for all time. "Greeks" as Latins initially known them from a Greek population lived around the area of Delphi temple that called themselves with that name and had settled in Sicily (in Latin Greacus from which the term Greek originated in English), Iliad also refer to Greeks as "Hellenes", "Graecians (Greeks)", "Argie (Argeans - Argos)", "Danaans" and "Achaeans". Other strong terms as "Romios" as Greeks called themselves from 324 AD until 1453 AD and later on up today, and Byzantines. At that time Greeks had large populations in Anatolia and Levant. So in the following decades the term Byzantine reserved to indicate that period of time in the history and this is how the term Byzantine remained.
The term Byzantine chosen to indicate that period in history, this is because at the time there (after the Greek Revolution 1821) was large Greek population still in the East (Anatolia and Levant) and few interest from the Italians of that time in terms for restoration. Greece at that time and true historically, sees Anatolia as its natural traditional lands with plans to liberate and free these regions. Of course advanced and backstepped in the following decades (eg. genocides against the Pontian and Ionian Greeks, as well as Armenian and Assyrian) on the same time with the results known and the modern Greek territory (1900-1921).
During 324 AD until 1453 AD Greeks referred for themselves as Romios which is similar or same to "Roman" in English. As the Eastern administrative of the Empire Hellenized on the same rate the term Greek fade away. During the transformation from the Olympian Gods to Christianity, the terms Greek or National (more common in levant) at the time refers to the religion, with many Emperors in Rome to declare themselves as Greeks to show their religious compass (Olympian Gods). On the same time mostly the Greeks in Anatolia turned to Christianity and as the term Greek refer to the old Gods, they start using the word Romios to identify themselves. Up today the Greeks in Anatolia and in the Middle East referred by the Arabs and by themselves as "Rum" (Romios, Roman). In the book of muhammadism (islam) there is a chapter on its own with the name Ar-Rum (for fast reference of the existence of the term from then to now). Today in Greece there are many Greeks who refer to themselves as Romios. For Greeks the term Romios, Ellinas (Hellenic) or even the english term Greek they are all mean the same and it is not a surprise or topic of debate, but it is common.
The Byzantine — Eastern Roman Empire — Greek Empire — Eastern Administrative Department of the Roman Empire from 324 AD until 1453 AD it is called simple as Roman Empire and its people Romios (singular), Romioi (plural). That's how the people of the empire called themselves and refer their empire. There was no other name but Roman Empire and Romios.