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The Middle-earth lies amidst the World, and is made of land and water; and its surface is the centre of the world from the confines of the upper Vaiya to the confines of the nether. Of old its fashion was thus. It was highest in the middle, and fell away on either side into vast valleys, but rose again in the East and West and again fell away to the chasm at its edges.

 

And the two valleys were filled with the primeval water, and the shores of these ancient seas were in the West the western highlands and the edge of the great land, and in the East the eastern highlands and the edge of the great land upon the other side. But at the North and South it did not fall away, and one could go by land from the uttermost South and the chasm of Ilmen to the uttermost North and the chasm of Ilmen.

 

^{The History of Middle-earth Volume 4, The Shaping of Middle-earth, The Ambarkanta, p. 238}

Then the Valar removed into the West and forsook the island; and upon the highland at the western side of the West Sea they piled great mountains, and behind them made the land of Valinor. [...]

 

For their further protection the Valar thrust away Middle-earth at the centre and crowded it eastward, so that it was bended, and the great sea of the West is very wide in the middle, the widest of all waters of the Earth. [...]

 

^{The History of Middle-earth Volume 4, The Shaping of Middle-earth, The Ambarkanta, pp. 238-239}

And the thrusting aside of the land caused also mountains to appear in four ranges, two in the Northland, and two in the Southland; and those in the North were the Blue Mountains in the West side, and the Red Mountains in the East side; and in the South were the Grey Mountains and the Yellow.

 

^{The History of Middle-earth Volume 4, The Shaping of Middle-earth, The Ambarkanta, pp. 238-239}

The Middle-earth lies amidst the World, and is made of land and water; and its surface is the centre of the world from the confines of the upper Vaiya to the confines of the nether. Of old its fashion was thus. It was highest in the middle, and fell away on either side into vast valleys, but rose again in the East and West and again fell away to the chasm at its edges.

And the two valleys were filled with the primeval water, and the shores of these ancient seas were in the West the western highlands and the edge of the great land, and in the East the eastern highlands and the edge of the great land upon the other side. But at the North and South it did not fall away, and one could go by land from the uttermost South and the chasm of Ilmen to the uttermost North and the chasm of Ilmen.

^{The History of Middle-earth Volume 4, The Shaping of Middle-earth, The Ambarkanta, p. 238}

Then the Valar removed into the West and forsook the island; and upon the highland at the western side of the West Sea they piled great mountains, and behind them made the land of Valinor. [...]

For their further protection the Valar thrust away Middle-earth at the centre and crowded it eastward, so that it was bended, and the great sea of the West is very wide in the middle, the widest of all waters of the Earth. [...]

^{The History of Middle-earth Volume 4, The Shaping of Middle-earth, The Ambarkanta, pp. 238-239}

And the thrusting aside of the land caused also mountains to appear in four ranges, two in the Northland, and two in the Southland; and those in the North were the Blue Mountains in the West side, and the Red Mountains in the East side; and in the South were the Grey Mountains and the Yellow.

^{The History of Middle-earth Volume 4, The Shaping of Middle-earth, The Ambarkanta, pp. 238-239}

^{Map of Arda before the destruction of the Pillars, The Atlas of Middle-earth, p. 2}

^{Map of Arda after the destruction of the Pillars and the creation of Valinor, The Atlas of Middle-earth, p. 2}

Side-note: Canonicity

It is often difficult to figure out what material was written the latest. Christopher Tolkien, in his History of Middle-earth series, never makes much of an effort to establish the "real story", that is, what Tolkien considered "real" on the day of his death.

It shows, I think, that even Tolkien himself was unsure about many things and that his mythology was in a constant state of change, sometimes with fundamental repercussions. Many important ideas were introduced late in his life, but without ever changing earlier texts.

The cosmological discussion above is based mostly on The History of Middle-earth Volume 4: The Shaping of Middle-earth. I know some of the terms were changed later on. For example, Ekkaia is used in The Silmarillion instead of Vaiya and its properties are somewhat different.

In short, what I wrote above is either correct, or was correct for a period of time. In any case, it gives a good idea of the origin of the name "Middle-earth".

Finally, near the end of the Second Age, Arda becomes round instead of flat and the island of Númenor is destroyed, but this event does not significantly change the geography of the continent.

In most of Tolkien's writings not concerned with cosmology, "Middle-earth" only refers to the central continent, not including the Seas. In colloquial usage, it can also refer to the Northwestern part of the continent where most of the stories about Elves and Men take place.

Finally, near the end of the Second Age, Arda becomes round instead of flat and the island of Númenor is destroyed. This event does not significantly change the geography of the main continent, but it has a major impact on the rest of the world: Aman and all the islands close to it are gone, the sea is now "bent" and goes around the world, and new lands appear.

The relation between the flat and round versions of the world is never made clear: is Vaiya still present? Are the lands still resting on it? Or is Arda now really "Earth-like" with real-world physics?

If we are now dealing with Earth, then "Middle-earth" becomes an archaic term because nothing is in the middle of anything anymore. In any case, Middle-earth has now lost an important feature that was technically a part of it: Tol Eressëa, the Ferry Island, which the far-sighted could see from the top of Meneltarma on Númenor.

In most of Tolkien's writings not concerned with cosmology, "Middle-earth" only refers to the central continent, not including the Seas. In colloquial usage, it can also refer to the Northwestern part of the continent where most of the stories about Elves and Men take place.

Side-note: Canonicity

It is often difficult to figure out what material was written the latest. Christopher Tolkien, in his History of Middle-earth series, never makes much of an effort to establish the "real story", that is, what Tolkien considered "real" on the day of his death.

It shows, I think, that even Tolkien himself was unsure about many things and that his mythology was in a constant state of change, sometimes with fundamental repercussions. Many important ideas were introduced late in his life, but without ever changing earlier texts.

The cosmological discussion above is based mostly on The History of Middle-earth Volume 4: The Shaping of Middle-earth. I know some of the terms were changed later on. For example, Ekkaia is used in The Silmarillion instead of Vaiya and its properties are somewhat different.

In short, what I wrote above is either correct, or was correct for a period of time. In any case, it gives a good idea of the origin of the name "Middle-earth".