"Middle East" is geographic-related, so Turkey is correctly listed among Middle Eastern Countries.
Arabic countries are such because of their language (and, by extension, culture), so you may have Arabic countries outside Middle East (Egypt) as well as non-arabic Countries within Middle East (like Turkey, as you correctly mentioned).
Boundaries are apparently vague, geographically speaking, as the US State Department defines the Middle East as only Egypt, Syria, Israel, Lebanon, Jordan, Iraq, Saudi Arabia, Kuwait, Bahrain, and Qatar. I understand that other countries are often lumped into the term, as well, including Iran, other northern African countries, and of course Turkey. Turks, however, do not consider themselves as "Middle Eastern", nor do most who live in the Middle East classify Turks as such.
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