The problem is, you defined "Roman" first using preconceived ideas and began assessing the Eastern Romans based on these ideas.

A Roman in the 1st century was primarily someone who spoke Latin, practiced Greco-Roman Paganism, and lived in Italy. A Roman in the 10th century was primarily someone who spoke Greek, practiced Orthodox Christianity, and lived in Asia Minor.

Both are unquestionably Romans. Both were recognized by their neighbors as Romans. In fact, up until the modern Greek nation state, most Greeks still identified as Roman, not Greek.

Ask an Arab, Turk, Armenian, or Persian what a Roman is, and they will describe the 10th century Roman. Ask a Western European or American, and they will describe the 1st century Roman. In truth, both are Romans, just from different time periods and geographies.