They were all called one thing once, Romans, when to be a Roman was the whitest thing you could do. For thousands of years.

Greece was the name given to the area by the Romans, at a time the people called themselves the Hellenes of Hellas.

The use of the term Hellenes fell out of favor among them as it was in the New Testament synonomous with Pagan.

Long after they finally start calling themselves Romans, long after they are divided into the Eastern Roman Empire, adopt Christianity and outlast the Western side by a thousand years, history decides that no, they were Byzantines.

Through Ottoman rule of the Eastern Roman Empire they were called Romans, and spoke Greek. During the independence movement of the 19th century they wanted to name the country Roma, Romania, something signaling their Roman ancestry, but historians of the former western empire considered them not be "pure" descendents of Rome. Strategically it was decided to use name for the ancient people of the area given by the Romans, Graeci, whose culture it was now claimed sowed the seeds of western civilization.

However not everyone got the message until surprisingly recently.

"Peter Charanis, born on [Lemnos] in 1908 and later a professor of Byzantine history at Rutgers University recounts when the island was occupied [in 1912] and Greek soldiers were sent to the villages and stationed themselves in the public squares. Some of the children ran to see what Greek soldiers looked like. ‘What are you looking at?’ one of them asked. ‘At Hellenes,’ the children replied. ‘Are you not Hellenes yourselves?’ a soldier retorted. ‘‘No, we are Romans.'"

One thing they never called themselves was white. It was an invented concept designed to divide and conquer the poor, allow for the concentration of wealth, and make slavery palatable to the local population that was now left out and could be doing the work. The average poor, local "white" person had no leverage in this situation, no comraderie with "white" immigrants they were now competing with, eg WASP's vs Irish, then vs Italians (who were oddly enough no longer sufficiently white or Roman).