Oh this is an interesting topic actually! The region we know as the Middle East has gone through many name changes, at least in modern history. In terms of the Europeans, Im afraid I cant speak to anything before the age of Imperialism, but the term "The Orient" was used first, around the 1500s until the 1600s or so, has remained in some usage since, and tended to refer to all of the East, from the Ottomans to China. This term would give way to the "Near East", which would last through much of the colonial period and obviously onward (The US State Department still calls it the "Near East Division". The "Nearness" of it would have a lot to do with the expansion of colonial holdings in India and Asia, which made the now called Middle East a nearer kind of East. Obviously this term is still in use as well, though the term Middle East really begins to be used in the early 1900s, though it was first used in 1850 by British Indian officials (who constantly squabbled over the term, and territory, with British Cairo officials). It was popularized by Alfred Thayer Mahan, an American and one of the great minds in geopolitics, who layed out the potential importance of the "Middle East" to the colonial powers as a coaling station on the way to India in a paper "The Persian Gulf and International Relations". The Middle East term stuck, and was generally the term in use until the term MENA began to appear in the early 9-11 period, which would incorporate North Africa as well, making more of a cultural than a geographic grouping, but any more discussion on this violates the 20 year rule.

As for the people of the Middle East, they have used, at least in the modern period, several terms to describe the region. First off, السرق الأوسط (Al-Sharq Al-Awsat, The Middle East), is widely used. As Edward Said pointed out in Orientalism, one of the facets of Orientalism is the native population accepting outside labels and terms, and thus the Middle East has become the most powerful geographic term in the region today, and it has been for some time. However, there are more indigenous geographic terms which still have value. بلاد الشام (Bilad Al-Sham, or often just Al-Sham) has a meaning of "Greater Syria", though it is usually translated as "Levant". This can refer to a potentially massive area, depending on who one is speaking to. Its generally accepted meaning is only the country of Syria, and occasionally Lebanon will be included. However, modern day Jordan and Palestine are included in this term as well, especially by those of a more Arab nationalist (or unfortunately as a more modern phenomena jihadist) bent.

For The Persian Gulf, several terms are used. It is popularly referred to as الخليج, (Al-Khaleej, or the gulf). However, its more proper name is جزيرة العرب (Jezeera Al-Arab, or "The Island of the Arabs") This name likely arose out of the agreed upon spreading out of the Arab peoples from the peninsula to the region following the creation of Islam, as well as the frequent comparisons made in the Arabic language between the sea and the desert.

As for Egypt, it has generally been called مصر (Misr, Egypt). This name has existed since the earliest documents I have seen and was in use at least since Muhammad's time. I have been told by an old professor that it was the term used by the Ancient Egyptians, which I would believe since large parts of Egyptian Arabic have uniquely Egyptian terms, though I couldn't verify that.

Anyway, I hope that was helpful, if anything needs clarification or expansion Id be happy to oblige. In terms of looking for more info on this, at least from the Western European naming history, Bernard Lewis' book "The Middle East and the West" covers this in detail. I know Lewis has in recent times become a bit of a political hack, (this book was written in 1965) but his work before 2001 is very good history.

It's Orta Doğu, "[the] Middle East", in Modern Turkish. For example, one of the best engineering schools in the Middle East is, Orta Doğu Teknik Ünversitesi, or in English, Middle East Technical University. But this is a borrowing directly from Western Languages.

The Near/Middle East is obviously the Near/Middle East from the perspective of Europe. In Arabic, too, people gave direction based names for things. Maghreb, the term used to refer to North Africa, come from the word Gharb, meaning "West", though I can only think of one other large region as strongly associated with an ordinal direction (see below). [in both cases, the final b would be a b in Arabic, but a p in Turkish/Ottoman; similarly, the Arabic g in Turkish/Ottoman is a silent "soft g"].

Ottoman geographers had a variety of names for the Ottoman Empire, some of my favorites include Devlet-i Âliye-yi Osmâniyye, the Sublime Ottoman State, and Memâlik-i Mahrûse, the Well Protected Realms. Ottoman geographer also subdivided the Ottoman Empire, the two most basic units were Rumeli, covering the Balkans all the way to Istanbul (so everything North and West of the Bosphorus), and Anadolu, Anatolia, covering Asia Minor. Both terms are originally derived from Greek, with Anadolu coming from the Greek Anatolía/ανατολία meaning the East Country, and Rumeli coming from Rum, Romans, which is how the Byzantines saw themselves. South of Anadolu was the Levant/Greater Syria/Bilâdü'ş-Şâm, which is Syria+Lebanon+Israel+Jordan and maybe the Sinai Peninsula. Levant is used in modern Turkish, but borrowing from Western languages, I believe it's originally Venetian but then borrowed into English and French, and from their into Turkish. I believe it has been used to refer to Latin Christians and sailors in the Ottoman Empire for a while, but not applied to a region in Turkish/Ottoman until more recently (in Western languages, it referred primarily or exclusively to the region). In Ottoman times, I believe the region would mostly have been called Şam, which you see in the news again because it's the S/L in ISIS/ISIL. This is confusing because in modern Turkish, this term refers mainly to the city of Damascus, though in Turkish for ISIS/ISIL is IŞİD: Irak ve Şam İslâm Devleti. Damascus's full name in Modern Arabic is Dimashqu sh-Shām, and is sometimes called ash-Shām colloquially in Arabic, as well.

West of Şam you have what we call today Mesopotamia, the land defined by the Tigris and the Euphrates, but I'm not actually sure what this was called in the Ottoman Era. Possibly "Irak", possibly something like the modern Arabic, "bilād al-rāfidayn", which means the land between the two rivers and has been a name for the region for more than a millenium. Likely both. The region that included both Şam and Mesopotamia was called Maşrek or Maşrik. Just as Magreb/Mağrip comes from gharb/garp, West, Mashriq/Maşrek comes from sharq/şark, East. Egypt is ambiguously included in this region, which generally refers to the lands between the Mediterranean and Iran. Egypt/Mısır/Misr, with its huge population, has generally sort of been considered of its own thing, though.

South of Maşrek you'd have Arabia/Arabistan/Ceziretü'l-arap [the Arabian Peninsula, literally the Island of the Arabs], in contemporary Turkish Arabistan or Arap Yarımadası [the contemporary way to say the Arabian Peninsula] and contemporary Arabic Jezeera Al-Arab, which is pronounced more or less the same as Ceziretü'l-arap; the C is a J sound and that t gets added to some Ottoman words because of which dialect Ottoman borrowed from. West of Maşrek you'd have the Iranian Plateau, but I've having trouble figuring out exactly what the Ottomans called it besides the Persian Empire. I'm pretty sure Persia/Persiya, but I'm having trouble confirming that right now.

Anyway, to my knowledge, there was no one historical term included the Levent, Anatolia, the Arabian Peninsula, Mesoptamia, Egypt, and Iran, with or without the Maghreb. You may be interested in this thing that a friend of mine, a history PhD student at Georgetown, made called 15 Maps That Don't Explain the Middle East at All, which is subtitled, "The region as it never was, could have been, and sort of is". Particularly maps 1, 13, 14. One thing he notes is that:

Over the past century, English terminology for the region has changed along with the geopolitical interests of British and American policymakers. In the 19th century, the British conceptually divided up what most of the world now considers the Middle East into the Near East (meaning the Balkans and the Eastern Mediterranean) and the Middle East (the region around Iran and the Persian Gulf, crucial for the defense of British-ruled India). The collapse of the Ottoman Empire after World War I made it easier for Europeans to consider as Western a large part of the former Near East, including the Balkans and sometimes Anatolia. It also brought much of the rest of the Near East under the same colonial authorities—the British—who were responsible for the Middle East. As a result, the term “Middle East” began to apply to colonies such as Palestine that were once considered in the Near East. Americans embraced this new usage of “Middle East” during World War II, and ran with it as the U.S. became politically involved in the region during the Cold War.

More recently, with the end of the Soviet Union, the term Middle East has again expanded in American usage to encompass Afghanistan, Pakistan, and Central Asian countries, as those places seemed to slide into a kind of Islamically themed, strategically relevant chaos.

[please forgive any mistakes in the Arabic or the Ottoman; the modern Turkish should be on point].

It's still "Middle East" in Chinese, Japanese, French, Hebrew, and probably more languages. In German and several Slavic languages they call it "Near East".

The actual "Near East" used to be the formal Ottoman Empire, but this term is no longer used.