"Fez" or "Tarboosh"?

Having recently got involved in an inconclusive argument on the subject the other day, I'd like to know which of these names is best when describing the hat worn in the Middle East.

Merriam-Webster defines a fez as a

brimless cone-shaped flat-crowned hat that usually has a tassel, is usually made of red felt, and is worn especially by men in eastern Mediterranean countries

and defines a tarboosh as a

red hat similar to the fez worn especially by Muslim men

but this doesn't get us very far. Does it matter what this hat is called, and if it does, what is the preferred term?

The Wikipedia article on this subject begins,

The fez (Turkish: fes, plural fezzes or fezes[1]), or tarboosh (Arabic: طربوش‎ / Egyptian Arabic pronunciation: [tˤɑɾˤˈbuːʃ], ALA-LC: ṭarbūsh) is a felt hat either in the shape of a red truncated cone or in the shape of a short cylinder made of kilim fabric.

A Google image search also confirms that these two words are used interchangeably for the same type of hat.

It should be noted that variations in the style and structure of traditional hats like the Fez are common and trends will often form geographically. For example in Morocco there is a common type of shoe that when purchased in the south of Morocco will have a different style than if purchased in the north.

Because the words will also divide themselves on geographic boundaries, based on both social pressures and the etymological differences between the words, it would not be surprising to find people who will use the words to refer to different styles of hats.

The Middle East is a big place. Fez is a Turkish word and tarboosh is Arabic. They refer to very similar hats, though there may be some region differences in design. In English, fez is used more than tarboosh.

"Tarboosh" seems to be the name for this type of hat that is used by Egyptians, in my experience. I am not sure about other parts of the Arabic-speaking world. "Fez" is the name widely recognised and understood by English speakers. Whether I would refer to it as a Fez or Tarboosh would depend on context - if the hat was being worn in Egypt, or I was talking to an Egyptian, I'd probably say Tarboosh. Otherwise, probably Fez.