I grew up distinguishing the Near East from the Middle East. There were no lines on the map, but the division seemed logical. Egypt, Libya, Lebanon, Syria, Jordan, and Palestine (Israel) were Near Eastern countries.
Saudi Arabia, Yemen, Iraq, Persia (Iran), Afghanistan, Kuwait, Bahrain, Oman, Qatar, Dubai, Abu Dhabi, and the other sultanates in south-east Arabia, were in the Middle East.
Then came India, Ceylon (Sri Lanka), Sikkim (now absorbed into India), Bhutan, (Pakistan, Bangladesh), Borneo, Sarawak, Brunei, Burma (Myanmar), Indonesia, Malaya, Nepal, and Tibet--all in the East, but not the Far East.
Finally, Japan, Formosa (Taiwan), China, the Philippines, Siam (Thailand), and parts of the Soviet Union, could properly be called the Far East.
I may have omitted a few past and present nations, but I'm sure you understand this breakdown, from a pre-WWll British perspective. But few (if anyone) in the US today speaks of the Near East. I checked this out recently, asking a well-educated American-born cotemporary what the expression "Near East" meant to her. It didn't resonate. So I am giving up my grouchy reaction when an announcer speaks of a country bordering on the Mediterranean (meaning in the middle of the world) as being in the "MIddle East".
Many Californians, and most of their parents or grandparents, came from the eastern side of the Mississippi. They continue to say "Back East" and "Out West". Since I reached California from a more distant shore, I have occasionally referred to the other coast as "Out East", but that hasn't caught on. Many a native Californian says "Back East", and I usually let it go! All I ask is that folk try to understand when I refer to (say) Lebanon as being part of the "Near East".