Does the term "middle east" cover a certain set of countries? Could I be misunderstood when using it?

I think all of the answers are great, but I will answer from a different perspective since I live in the so-called Middle East.

As some said, There is no official definition of middle east. But there are several definitions made for different purposes.

For example, As an Arab we Arabs usually use Middle East to refer to Arab Countries. Regarding Israel (we Arabs don’t use Israel, we call it Palestine), it is considered to be an Arab country since we believe it’s an Arab land. Regarding Iran, We don’t consider it as a middle east country while Iran is only few kilometers away but we are totally different, Language, Food, Dress, culture and even different type of Islam.

On the other hand, non-Arabic media use the term Middle East to refer to the Islamic world in general (or what they call the Greater Middle East). This term is used in the western-media especially when the term terrorism is available, When used for this purpose Iran is included and Israel is excluded.

About Iranians (Persians) and Arabs, We actually have a very long history of wars and conflicts so we prefer not be included in the same category not only in the middle east issue but also in religion issues (since Iranians are mostly Shia and Arabs are mostly Sunnah). These problems started a long time ago specially when the small Arab army defeated the great Persian army in the year 636 in the Battle of al-Qādisiyyah. Right now Iran is occupying the Arab land of Arabstan and three small islands which belong to the United Arab Emirates. Another issue is the Arabian-Persian Gulf conflict.

Regarding Palestine (A.K.A. Israel), as @littleadv said, Arabs and Jews are actually cousins, Language, dress, food, culture even our religions are somehow close but both of us do not like to be included in the same category. While Jews in Palestine are considered as part of Europe in some cases, Arabs in Palestine are considered to be from the middle east!!

Bottom line: Middle East is a term that can be used in two identical sentences said by two different people resulting in two different meanings!

UPDATE:
Some have taken offence at my words when I said “Palestine (A.K.A. Israel). This is not a personal belief nor a political issue, this is a fact. Check this this Wikipedia page:

Palestine (Arabic: فلسطين‎ Filasṭīn, Falasṭīn, Filisṭīn; Greek: Παλαιστίνη, Palaistinē; Latin: Palaestina; Hebrew: פלשתינה Palestina) is a conventional name, among others, used to describe the geographic region between the Mediterranean Sea and the Jordan River, and various adjoining lands.[1][2][3][4][5][6] The region is also known as the Land of Israel (Hebrew: ארץ־ישראל Eretz-Yisra’el)

Middle East is a geographical area east of the Mediterranean sea and west of India. Sometimes the “stans” are included (Afghanistan and Pakistan) but more often they’re not because when the term was originally coined they were part of British India.

As mentioned in the comments, North African countries are considered “Middle East” in some cases, because they are such geo-politically. On the other hand, Israel is considered “European” even though it’s nowhere near Europe, in some cases, especially for geo-politically based affiliations such as UN groupings or sports affiliations.

Iran and Israel both are definitely middle-eastern countries, geographically.

Semitic people are people originally from the Middle East, but don’t have to be so now. Jews are considered Semitic people (hence the most prominent usage of “antisemitism” being essentially anti-jewish behavior), but most Jews are outside the Middle East area (approximately half of the world Jewish population have Israeli citizenship, of which many live outside of Israel).

Arabs are also Semitic people. Persians are not. Arabic and Hebrew are closely related (even though the alphabets are significantly different at first glance), Arabic and Persian are not related (even though they share the alphabet and many words).

Muslim people are people who are of Muslim faith. Don’t have to be Semitic at all and not all Semitic people are Muslim (there are Jews who are Jewish, Arabs who are Christian, and Persians, Indians, Indonesians and Europeans who are not Semitic, but are Muslim).

So these are three different terms (geographical, ethnic and religious), but many people can be described by combination of those.

Are all countries between the “near east” and the “far east” then
“middle eastern” countries?

I always thought Middle East and Near East are mostly synonyms. (For me, this is probably influenced by the fact that the Finnish word for Middle East is Lähi-itä, literally Near East.)

Even if we stick to English terms, Wikipedia tends to agree (emphasis mine):

The term Near East was in use exclusively during the 2nd half of the
19th century. In the 1st half of the 20th century it began to share
the geographical stage with the term, Middle East. Since then Near
East and Middle East have been approximately synonymous. Near East is
used in some historic contexts and Middle East in others with no major
semantic difference.

So, as to your other questions:

Are Iran and Israel “middle eastern” countries?

In common modern usage (at least modern “Western” usage), yes, Israel and Iran certainly are Middle Eastern countries. When there’s something on the news about the Middle East crisis, etc, quite often it is somehow related to these countries (among others), isn’t it?

^{“Countries most commonly considered as part of Middle East”, according to Finnish-language Wikipedia. Image from Wikimedia Commons.}

Or is “middle eastern” supposed to be a synonym for “Arabic”?

No, I don’t think so; never heard of such idea before. I think Middle East is first and foremost a geographical term.

Of course, there is no “official” definition for these terms, and anyone is free to use them as they wish. It would likely be futile trying to arrive at a definition everyone agrees upon, especially given the contentious recent history of the area.