I'm from Malta and the word Ottoman is never used. We (including historians) say "In 1565 Malta was attacked by the Turks". Shouldn't the word Ottoman be used instead?
Yes, I would say it is more accurate to use the term "Ottomans" rather than "Turks." For essentially all of its history, the Ottoman Empire was a multi-ethnic empire and brought in people from almost all subject ethnicities into the political and military class. The grand vizier of the most famous Ottoman Sultan, Sulayman the Magnificent (i-Kanuni), was Pargali Ibrahim Pasha, who was of Greek origin and taken as a boy via the devshirme. The Koprulu dynasty, which provided six grand viziers, was of Albanian origin. Even those who were ostensibly Turks, such as the grand vizier Karamani Mehmed Pasha, traced their descent to non-Turkish brackgrounds, in this case the Persian mystic Jalal al-Din Rumi.^1 The Kurdish scholar Idris Bitlisi was a confident of Selim I, his biographer, and played a key role in expanding Ottoman sovereignty over southeastern Anatolia. Such examples are too long to list. Even in the late 19th and early 20th century, when Turkish nationalism emerged as a political force among the Ottoman elite, many of the so-called Young Turks were not actually Turks. An example would be the Kurdish intellectual Dr. Abdullah Cevdet.
That being said, Europeans writing about the Ottoman Empire during its existence very commonly used the term "Turks" to describe its political and military ruling class and "Turkey" to describe the empire itself. Of course, they also described someone who converted to Islam as "turning Turk." As for the Muslim world historically, I generally see terms like dawlat al-'Uthmaniyya (i.e. the Ottoman state) being used to refer to the Ottomans. However, I have even found cases in which non-European Muslims used the term "Turks" to refer to the Ottomans, such as the Qasimi Imams of Yemen.^2
This is a more complicated question than it appears on the surface. There are reasons to use the word "Turks," since the progenitors of the empire were Turkic people. Their language was also Turkic, and in the early days of the empire, nothing about that was seen as problematic. As a sort of a shorthand, when talking about a particular time period, or talking about a particular event like the siege of Malta, using the word "Turks" isn't likely to create any confusion.
On the other hand, the Ottoman Empire very early on became a European-focused empire. After Anatolia, their territorial gains began to pile up in Europe. In the 14th century, they conquered much of Anatolia and most of the Balkan peninsula. Tellingly, the capital was quickly moved to Adrianople (Edirne), in Europe, and it would remain in Europe for the rest of the empire's existence. In the 15th century, they consolidated their gains in Anatolia, took Constantinople and made it the capital, and then conquered the rest of Balkans, Wallachia (now the southern part of Romania), and parts of Hungarian territory, as well as parts of modern Ukraine and Russia, specifically the Crimean peninsula and the areas surrounding the Azov. It was really not until the 16th century that they headed into the Middle East, Egypt, and North Africa.
On top of this, if you look at the makeup of the Ottoman armies in the Renaissance, you'll find quite a lot of Europeans. Slavs, Albanians, Italians, Greeks, Magyars, etc. They all fought alongside the Ottomans, some because they had to and some because they wanted to. Ethnically these people were not Turkic, nor were they all Turkic speaking to any great extent, although there were dragomans to help with translation. In this same vein, if you look at the viziers, generals, and admirals around the time of Selim II (to use a useful example, because he began his reign directly after the aforementioned siege of Malta), you'll find quite a lot of Europeans. Mehmet Sokollu and Lala Mustafa Pasa were Bosnian. Sinan Pasa and Piyale Pasa were Croat. Semsi Pasa and Koca Sinan Pasa were Albanian. Kilic Ali Pasa was Italian! (This is not to say that ethnic Turks - or others - were not present, but to give some idea of the diversity of non-Turkic people holding enormous power in the Ottoman Empire.)
Even the Ottoman sultans themselves, after a time, were not "purely" Turkish. Mehmet II's mother was a former slave girl of possibly Italian or Serbian descent. Selim II's mother was Polish. Mehmet IV (who besieged Vienna) had a Ukranian mother. Again, not to say that Turks weren't present or that bi-racial children can't be authentically Turkish, but this ought to give some idea of the melting pot that was the Ottoman Empire. Calling every Ottoman a Turk just isn't going to be accurate.
Additionally, there is even some evidence of anti-Turkism in the late Ottoman Empire. Even the word "Turk" came with baggage - it was largely used in a disparaging way when talking about country farmers in Anatolia. Many of the higher-ups in the late Ottoman Empire wanted to nothing to do with the idea that the Ottomans were Turks. That was offensive to them, and they much preferred to be called Ottomans. There is also anti-Turkism present in the idiomatic constructions of many of the languages of the various countries that were formerly ruled by the Ottomans.
Finally, I think it's useful in some cases to specify "Ottoman" over "Turk", to avoid the appearance of attribution error. The Armenian Genocide? It wasn't "the Turks," as if having Turkic ethnicity automatically makes you complicit. It's the same reason why, when talking about the Holocaust, I always make it a point to say "the Nazis" did it, and not "the Germans." There's nothing particular to being German that would lead to a Holocaust.
To sum up, my personal preference is to specify, but I understand the use of the word "Turks" as shorthand for "subjects and rulers of the Ottoman Empire" when it won't lead to confusion and/or is more poetic or prosaic. Ultimately, however, if there's confusion or if it's likely to cause the listener to falsely attribute something negative to a race/ethnicity as opposed to the government policy of a largely-European nation, the term "Ottoman" is just better.
To add to, rather than contradict, the people who have already written:
The word 'Ottoman' is a European bastardisation of an Ottoman title (referring to Osman). To that end, it seems like an odd title to use. But the Ottoman empire was dynastic: it wasn't by any means mainly made up of Turks. It wasn't even majority muslim until the capture of Cairo in the early sixteenth century. So I use Ottoman because this emphasises the dynastic rather than the supposed ethnic foundation of the empire.