I don't have my books to hand, but I'll try and explain better.
Arab (a word first apearing in the 9th c. BC) originaly reffered to the Bedouins and the language of the Arabian Peninsula. In the first few decades after the Conquest it ment the ruling elite, who were Muslims from Arabia. Non-Muslims were Dhimmi and those who converted- and supposedly enjoyed equal status with the Arabs- Mawali.
Due to economic and cultural changes, the reign of Turkish dynasties, the widspread use of Arabic in the towns, and scores of demobilised Arabs who mingled with the locals, the Arab tribal castes lost their predominace and the word Arab reverted back to the sense of a nomad. The rest called themselves Muslims, or to differentiate from Turk and Persian, Children of the Arabs. The Mawalis to all intents and purposes became equal.
By the end of the 19th Century, the European idea of a nation being characterised by a common homeland, langauge, character and political aspirations was adopted by the intellectual movements in the Mid-East and that sense has come down to today.
Egyptians are Arabs in the modern, nationalist sense of the word, while Copt is now restricted to the Chrisitan minority.