|Sultan Selim III exchanging eid greetings in front of the Gate Of Felicity**|
Ottoman scholars and intellectuals largely used to designate themselves Rumi in a sense that we can consider to be "ethnic": this term, meaning Roman or Byzantine in early classical Arabic, carried the sense of Anatolian-Rumelian in Ottoman Turkish. Nonetheless, it was also not denied that the majority of the Rumis spoke the "Turkish language" (lisan-ı Türkî or Türk dili), at least as a lingua franca, and were related to the Türk communities, who were mostly either nomads or villagers living mainly by raising livestock.
It is quite easy to find on the internet examples of the senses in which the Europeans used such terms as "the Turkish Empire" or "Turkey". For instance, let us have a look at this map: John Speed 1626. Here the term "the Turkish Empire" is used to define the map. According to the information below it, it was made in 1626 in London.
We will now see another map. Titled "Turkey in Asia", this map comprises such Arab-majority countries as Syria, Iraq and the western Arabian peninsula. The year of the drawing is 1900: Turkey in Asia.
As observed on these two maps, Turkey meant the whole Ottoman domain for Europeans until only a century ago. Besides, a Turk was defined not as a Turkish speaker but as a Muslim, a believer in Islam, even though from the late 19th century onward, it would start being restricted to native Turkish speakers. As a matter of fact, the idiom "to turn Turk", that is, "to become Turkish" used to mean "to become Muslim" in English. There were not any connotations relating to the Turkish language here. For example, if someone converted to Islam in Tunisia in the presence of Arab Muslims, even this would be termed to be a case of turning Turk.
Then, even such words as we think to be the most basic and simple can turn out to have had such complex and surprising historical meanings until recent times, right? Although change in the meanings of words over time is basically normal, we should not underestimate the too rapid and abnormal change, the swing, induced by the fast modernization and the currents of nationalism that have been going on since especially the late 19th century.
** The image has been borrowed from Wikipedia/Wikimedia; it is in public domain. It was painted by Kapıdağlı Konstantin, an Ottoman painter of the period.