English does not care what terms people use for themselves, or what terms they do not like to use for themselves, it is what it is. Turkey is in the Middle East, Cyprus is in the Middle East, they are in it whether or not they like it.
Near East and Middle East refer to the same thing, so saying Turkey is in the Near East is just the same as saying Turkey is in Western Asia, it is still the Middle East, no matter what you call it.