I can't say much of the early modern period, my specialties are antiquity and the Middle Ages. But it is true that the elite of the Byzantine Empire hated being called anything other than Roman, and they would strongly disapprove of us calling them Byzantine. However, it was not a response to accusations of being pagans, it was because they were the direct continuation of the Roman Empire, so they continued to call themselves Romans. The Alexiad, a work by the Byzantine princess Anna Comnena in the 12th century, shows that the Byzantine elite still firmly viewed the world in Roman terms - she even referred to the French as Gauls and Kelts.

However, snide European authors would sometimes refer to the Byzantine Emperor as 'King of the Greeks' to mock their diminished status in the world. Very few European authors acknowledged the continuity between the Roman Empire and the Byzantine Empire, partly out of ignorance and partly because the Holy Roman Empire held that position in the west (as an aside, the Byzantines never recognised the Holy Roman Emperors as legitimate counterparts and it was a constant source of tension). The most drastic incident occured in 968 when a letter from Pope John XIII, which called Emperor Nicephorus Phocas 'Emperor of the Greeks' and not 'Emperor of the Romans', caused a diplomatic crisis and the resumption of military clashes between Byzantine and Holy Roman forces in Italy, a bitter account of which was written by the diplomat and bishop Liutprand of Cremona. The text is called Relatio de Legatione Constantinopolitana and is here if you want to read it in full: https://sourcebooks.fordham.edu/source/liudprand1.asp