The Eastern Empire never spoke Latin. There was always a cultural divide.
In Egypt for example the Ptolemaic dynasty spoke Greek (Cleopatra was the first to even learn Egyptian) and it was still widely used in Roman times.
"The Western Empire never spoke Latin."
Not to mention that according to the article, that eastern Roman Empire also called itself Romania.
Romania's longing to Roman history is mostly a nation building effort. Romania was a buffer territory between the Ottomans, Habsburgs (austria-hungary) and Huns for many centuries. It was so hard to get people to live there, because there would always come an army to torch everything to the ground. They lured Germans to live there and civilize the country in return for city rights.
They considered themselves Roman up until Constantinople fell to Mehmed II. And even then, the Ottoman Turks took the title Kayser-e-Rum, literally "Caesar of Rome." And even before them, as the Muslims first expanded out of Mecca and Medina, they referred to the Byzantines as "Rum" or "Rome."
Not as much, especially after the first 4 centuries. They claimed a different heritage then (the local one).
Besides the western barbarians that took down Rome proper also called themselves Roman ("the Holy Roman empire") for "legitimacy" purposes (as a claim to the expanses of the empire). The same reason Mohamed the conqueror adopted the "Ceasar of Rome" title.
- They were no longer properly Roman after they stopped speaking Latin, which they did almost immediately because majority of territories remaining under their control after initial Arab conquest were Greek-speaking.
- Written cultural heritage of Byzantine, unlike Roman, is negligible, mainly because they intentionally tried to be as un-original as possible, which is a paradox: they valued not writers or poets who could write new stuff that sounded fresh, but those who could imitate particular ancient authors's style as precisely as possible.