A number of these are subtly different or used to describe different phenomena. The Fertile Crescent might be in the Middle East but its a term used to describe the ancient civilizations the popped up in the crescent shaped region from Mesopotamia to the Nile. Although it describes a geographical area I've never seen that term used in any other historical-political context (e.g. I have never come across anyone describing the Arab heartland of the modern Ottoman empire as being in the Fertile Crescent.)

Then there are language differences. Levant comes from French and ultimately Latin meaning "rising" i.e. "Land of the Rising Sun." In that meaning it's identical to the Arabic term "Mashriq." Mesopotamia is Greek meaning "Between the Rivers," i.e. the Tigris and the Euphrates. The corresponding Arabic term is "Al Rafidain."

In addition to the list you provided we might also add terms like Anatolia, Asia Minor, Outremer, Bilad ash-Sham, Al Jazira, Arabia, the Orient (which in Britain typically refers to the Near East while in the US it typically refers to the Far East), Greater Syria, Greater Israel, Greater Persia, Greater Armenia, the Former Ottoman Territories, the Arab World, the Arab Middle East, MENA (Middle East and North Africa), Islamdom, and on and on and on.

All of these terms are useful in illustrating different political or historical facts and territories even if they overlap. Some are more precise than others. Some are better known than others. Some are older, some are newer, some are more or less politically charged. Some are regarded as being more or less Eurocentric or Imperialistic.

Unfortunately I don't think there's going to be a great answer to your question except to say that linguistically and politically this region has been a focus of human civilization for thousands of years and over time it's acquired quite a lot of names. These names do, however, tend to have different meanings, connotations and uses, so they aren't really as redundant as you might think.

edit: Typo

Edit 2: one example that I think is interesting is when the British were basically dividing up the former Ottoman Empire at the end of World War I. The men (and one woman, Gertrude Bell) who took on this task tended to come from similar public school, Oxbridge backgrounds, usually with a heavy emphasis on classical languages. There was a strong push on the part of these classically trained administrators to name the modern state of Iraq as "Mesopotamia" but Faisal, the other Arabs, and a few of the British on the ground thought it would be totally ridiculous to impose a Greek name on an Arab country. They ultimately won the argument.

Most of these terms don't exactly overlap, though they refer to regions which are close to each other.

Levant is usually used to refer specifically to the Mediterranean coast of the Middle East--Israel, Syria, etc. Sometimes Iraq is included, but not always.

Mesopotamia refers specifically to the area around the Tigris and Euphrates rivers, the land of modern-day Iraq down to the Persian Gulf. The Mediterranean coast is not included in Mesopotamia.

The Fertile Crescent is a broader term which includes both Mesopotamia and the Levant, as well as the Nile Delta.

The Eastern Mediterranean basically refers to the Levant, Asia Minor, Egypt, etc. So a broader way to refer to the whole Mediterranean coastal area.

As far as I know, Near East and Middle East are equivalent, and Near East is just a more old-fashioned term.