So as I understand it, there used to be three distinct terms to refer to parts of Asia (apart from the Russian part): Near-east, Middle-east and Far-east.
Middle-east refered to what is now Iran, Pakistan and India. Far-east to anything east of that. And Near-east to The Levant, Mesopotamia and Turkey.
Nowadays though, I see the term 'Near-east' rarely used. And 'Middle-east' seems to cover both the areas of the old 'Near-east' and 'Middle-east'?
When and why did this happen?
There are other things to add to your question but you might want to have a look at the answers to this question from myself and /u/textandtrowel on the usage of these terms: https://www.reddit.com/r/AskHistorians/comments/8d5v94/short_answers_to_simple_questions_april_18_2018/dxl7cbj/
Happy to try to take any follow ups if that doesn’t cover it, but the basic answer for “when” is by and large the first part of the 20th century, and the “why” is (to massively oversimplify) a decreased relevance of the “nearer” bits (namely, the Balkans) and a greater relevance on the “middle” bits for geopolitics.
That being said, while “near east” has fallen out of favor among the general public, it’s still present for specialists, even if only as an anachronism. The US State Department’s “Near East Division” comes to mind.