We are here to help so need to be confused! So first some basics:
- What is nowadays referred to as "Eastern Anatolia" is a term that was invented at the end of the 19th century and only gained recognition after the Armenian Genocide thanks to Turkish efforts. The reason is simple: most of "Eastern Anatolia" is in fact the Western portion of Armenian Highlands and has been called so for more than two millennia. Now, the borders of Armenian Highlands aren't fixed so in the south it transitions into Mesopotamia and in the north into Caucasus and Pontus, in the west into Anatolia (specifically Cappadocia). The fact that many people use Mesopotamia to refer to the historic region but also use "Eastern Anatolia" in tandem shows how complete the eradication of the term "Armenian Highlands" has been. So, when you say "Eastern Anatolia" in a historical context and also mention Armenians and Kurds, I assume you are talking about mainly Western Armenia, Kurdistan (which started to be used relatively later on) and perhaps also Pontus. Or maybe you are referring to Cappadocia? As Eastern Anatolia in the Middle Ages referred to mainly Cappadocia. But I assume you aren't talking about it.
Now for the question itself:
- By the Middle Ages, Armenian Highlands had been predominantly Armenian for quite some time. In fact, Strabo mentions that in the 1st century BC people in Armenia spoke one language, which is nowadays regarded as being Armenian. That is not to say that there were no other people living there: there was a large Jewish diaspora and especially in border regions of Armenia population was quite mixed or even non-Armenian. Especially in the southern regions of Western Armenia, which bordered Mesopotamia, there was a strong Syriac presence already in Antiquity. And there some evidence that proto-Kurdish people already lived in those border regions between Mesopotamia and southern Armenia in Antiquity.
- After the Arab conquest and the Armenian rebellions in the middle of the 8th century, the Ummayad Caliphate sponsored a large scale Arab migration into key Armenian regions, which resulted in some Arab enclaves in major cities and especially in the southern border regions and around Lake Van. But they still constituted a minority and the region was predominantly Armenian.
- By the middle of the 11th century, the Eastern Roman Empire had conquered most of Western Armenia and the Arab presence was mostly undone. And of course in 1071 Seljuk Turks won at Manzikert. Now, at that point, Armenians still constituted a majority, but there were important minorities like Kurds in some southern regions as well (which would start to be called Kurdistan later on).
- Kurdish tribes had started to move from Mesopotamia (so approximately south of Lake Van) northwards into Armenia proper and with the arrival of Seljuks, this process accelerated but only reached massive proportions in the early 16th century when the Ottomans allowed them to settle in Western Armenia to have a reliable warrior base in the region against Safavid Iran.
- Pontus had every mixed population: the majority were Hellenized and Greek-speaking locals. Who were the locals? Likely mainly Laz people and Armenians of Armenia Minor. This composition stayed basically the same from the early days of Classical Antiquity into the Middle Ages until the arrival of Turkish tribes.
- As for Mesopotamia, it is a bit harder: most of Mesopotamia was populated by various Semitic people, mainly Arabs and Assyrians, with a notable Jewish presence and of course Iranian speaking people in Northern Mesopotamia (proto-Kurds). By the 11th century, Kurdish principalities had risen substantially in importance in Northern Mesopotamia, where the population was in likelihood mainly composed of Kurds and Arabs. Because of Armenian migrations, there was also a noticeable Armenian presence in some regions of Northern Mesopotamia (for example Edessa) at that point.
- So to put it simply, by the arrival of Turkish people, Western Armenia (aka "Eastern Anatolia") was predominantly Armenian, with notable Kurdish presence in southern regions (Kurdistan). Mesopotamia was mainly inhabited in the south by Arabs, and in the north by Kurds, with a notable presence of Assyrians and to a lesser degree Armenians.
- As a final note: it is very hard to understand the population dynamics after the Seljuk arrival, but it sometimes assumed that Armenians constituted an absolute majority in Western Armenia until about the 15th century when the devastating campaigns of Tamerlane delivered a massive blow to Armenians. Centuries of nomadic Turkic and Mongolic invasions and migrations, widespread destruction and massacres, and the northward migration of Kurdish tribes resulted in Armenians becoming a minority in some regions of Western Armenia after the 15th/16th centuries. This trend would continue until the 20th century and would result in a very difficult ethnic composition of Western Armenia. Mesopotamia generally stayed the same, with the notable exception of Assyrians, who were dealt a major blow by the massacres conducted by Tamerlane as well and the settlement of some Turkoman tribes in northern Mesopotamia. And in Pontus Turkification and Turkish migrations (mainly after the 15th century, when the Empire of Trebizond fell) gradually resulted in Turkic speaking Muslim population to become the dominant element.
Hopefully, I have explained in a coherent manner and if you have any follow-up questions, I would be glad to answer!