At what point did Rome or the Eastern Roman Empire become known by people outside of its borders as a Greek state, rather than Roman. I know the majority of Byzantium's citizens considered themselves Roman, but at what point did Greek become the language of administration rather than Latin?
In 620 C.E., the Greek language was made the official language of the Byzantine Empire by the Emperor Heraclius. Prior to this, Latin remained the official language of administration, and bureaucrats and military officers needed to be able to read and write it. However, being the dominant language of the eastern Mediterranean since the Hellenistic period, Greek was still used in official conversations and everyday administrative records, and was the more prestigious literary language. Under Theodosius I, the staff of the imperial library at Constantinople had three Latin and four Greek copyists, and Theodosius II created chairs for thirteen Latin and fifteen Greek professors. You could say that Greek had always been a language of administration in the ERE, just not the official one until Hercalius' reforms.
With the brief loss of Egypt and the Levant to Sassanian Persia around 620 C.E. and their permanent losses to the new Arabian Caliphate, the ERE was reduced to core territories centered around Greece and Anatolia, almost exclusively Greek-speaking, and at this point Greek essentially became the empire's sole language. With the theme system, Constans II began to use Greek names for the Roman armies, rather than Latin ones seen in the legions.
I'm not too sure about Western European perceptions of the Byzantines as a Greek state, particularly during the Crusades, so I would be really interested to learn about it if anyone could shed light on the topic. IIRC, the Byzantine Emperor was often mocked by Western Europeans as being called the "Emperor of the Greeks", and the Byzantine empire as being the "Empire of the Greeks", separating it from the Roman prestige so revered throughout the Mediterranean and medieval European world. That is not to say that the emperor wasn't the ruler of the Greeks, but it ignored the emperor's position the Christian Roman succession that began with Constantine I.
In the Palaiologan period, some Byzantines began to use the title "Hellene", or Greek, as meaning an inhabitant of Greece. Prior to this, being called a Greek was associated with paganism and as un-Roman and un-Christian. A minority of Byzantine scholars began to take pride in being descended from the pagan Greeks besides being Roman and Christian, though I would not think there would be some sort of national Greek consciousness. The reign of John VI Kantakouzenos saw a revival of learning, and Greek classics were copied and studied, and one scholar, Demetrius Triclinius, reconstructed the meters of ancient Greek poetry, among many others who contributed to classical learning in the renewed interest of the classics. There was at least an understanding and realization of the Greek past as Byzantium became more ethnically homogenized due to territorial losses.
Source: Warren Treadgold, A History of the Byzantine State and Society