While "Turkey" only came into existence as a nation state after WWI it was extremely common, at least in English to refer to them as "the Turks" and had been for several centuries. Thus, the equivalent of the German "Fritz" or the English "Tommy" was "Johnny Turk" or sometimes more accurately "Mustafa." So if you read Churchill's history of the First World War, for instance, he constantly refers to "The Turks." You can also make a relatively reasonable case for that being accurate by the time WWI broke out. After the institution of Turkification policies at the end of the 19th century and the Young Turk revolution at the start of the 20th I think you could make a case for the Ottoman Empire actually being "Turkey" shackled to an Arab hinterland by the Ottoman sultanate/caliphate. That's ultimately why Ataturk's view of Turkey isn't revanchist.

In any event, I think the key thing here is usage. Dan Carlin does it because people like Churchill did it. As a technical point of usage it doesn't matter very much, by the 19th century if a historian says "Turkey" there are only so many possible meanings to that, and contextually it inevitably means either the Ottoman Empire as a whole or its ethnically Turkish Anatolian heartland.

The Ottoman Empire was a multi-ethnic empire by the Osmanli dynasty at the head of Turkish tribe. The Ottoman Empire would be somewhat analogous to the Habsburg Empire (which was also a multi-ethnic empire centered in modern day Austria ruled by the Habsburg dynasty). Before the rise of nationism movements that were popular in the 19th and 20th centuries, the Ottomans were not organized along ethnic or national lines such as the predominance of Turkish people or Turkish customs and culture. It would have disenfranchised the Arabs, Egyptians, Mesopotamians, Berbers, Libyans, Armenians, Greeks, Bulgars, Romanians, Bosnians, Serbians... Etc. In it's later years, the Ottoman Empire began to actively pursue the creation of an "Ottoman" identity to counter the growing divide created by nationalist movements based on ethnic and cultural lines. In the late 19th and 20th centuries, the Ottomans were plagued by the Balkan Wars were nations such as Bulgaria, Greece, Bosnia, Serbia and others started to break away from the empire. Similar movements occurred in Egypt and Armenia. As the 20th century started and WW1 drew close, it became harder and harder to deny that the Ottomans were not a Turko-centric empire. In fact, the Allies exploited Arab nationalism and other minority nationalist movements to undermine and revolt against the Ottomans in WW1. It was after significant territorial losses encompassing most of it's minority groups that the Ottoman Empire was replaced by the Turkish Republic.

Edit: I wanted to add that the Turks as an ethnic group held a lot of power in the region long before the founding of the Ottoman Empire. Many Turkish rulers in control of Syria and the Levant participated in the Crusades. Prior centuries, much of the middle east was dominated by the Seljuk Turkish Empire from Anatolia(modern day Turkey) to parts of Iran/Persia.