I've read that the (greek) citizens of the late (Eastern, Byzantine) Roman Empire, aswell as romanians in Wallachia, Transylvania and Moldova used the term Romania or such variations when talking about the realm or area they lived in, refering to themselves as romans. This continued with the greeks until the greek declaration of independence and the roman identity was replaced by a more nationalistic hellenic identity, aswell as the term Romania surviving until today, becoming the name of the nationstate of todays romanians.
Now I wonder, when did this term come into existence? Was it ever used by the romans before the split into east and west Rome to refer ot the entire realm?
The term Imperium Romanum (Roman rule) became official during/after the reign of Augustus (27BC-14AD). Around 200AD all inhabitants of the Empire were granted the Roman citizenship, thus they all were Romans, regardless of their ethnicity, which actually didn't even exist the way it is now defined. It is therefore very hard to speculate about a Roman identity other than "Roman citizenship".
Anyway, the term "Romania" as such was not used at all during the Roman Empire and appears to be a later, medieval creation.
During the Roman Empire the "Roman realm" was equivalent with the "Roman rule" (Imperium Romanum), which was inahabitated by "Romani" (Romans) in the sense of Roman citizens, inhabitants of the City of Rome.
After the split into East and West at end of the 3ird century and the switch from Latin to Greek as official language of the Eastern Empire in the 7th century, the Greek term "Rhomania" may have appeared in what has later been called the "Byzantine Empire".
To answer your question, the term "Romania" is a medieval coinage in both West and East: in the West it designated the Romance speaking and religious (catholic) realm claiming a Roman heritage (from Portugal to Italy), in the East the term "Rhomania" was just another word for Basileia tōn Rhōmaiōn, the Roman rule, Imperium Romanum.
The Greek speaking "Byzantins" considered themselves naturally Romans, Rhomaioi living in Rhomania. These endonymes (Rhomaioi, Rhomania) were political: those people considered themselves members of the Roman rule.
The term Romanian/Romania designating the modern state of Romania has the same roots yet a completely different history.
Many Italian and some German and French late medieval records (XVIth century and further) document that the inhabitants of today Romania called themselves "Romans" in their own language, that is "români/rumâni".
As for "Romania" as a land name, the first known record dates from 1521, where Wallachia is called in Romanian language "Romanian Land" (Tara Romaneasca). The document is known as "Neacsu's Letter.
There is however a notable difference in the meaning of the self-designation "Roman" between Byzantins and medieval Romanians, called Vlachs or Wallachians by foreigners.
While the Byzantin meaning of "Rhomaioi" is political, the Romanian meaning of "român” is foremost linguistical. For the medieval Romanian, being "Roman" meant just speaking "Roman", a Romance language. A certain conscience of a "Roman" origin and heritage among Romanians is also attested by some medieval foreign documents, but I doubt this could be true outside the narrow circle of the lettered.
To make things even more complex we have the Balcan Romance speaking people of the Aromanians (Armâni, in their own language), who also understood themselves as "Romans".
Actually, the Roman name generated more than one designation, to mention Romagna (region), Rumantsch (language), etc.