Wikipedia has a pretty comprehensive list of the names used for the Ottoman Empire at different periods and in various languages. At the end of the article there's also a chronological list of links to historic maps using the alternative names of the Ottoman Empire.
Since you are mostly interested in diplomacy and official writing, I also looked for a few notable international or bilateral pacts and treaties. The period of the Ottoman Empire I'm mostly familiar with is during and after the Greek War of Independence, so I focused my search on documents after 1821. Nevertheless, I think my findings sufficiently show that a variety of names was used, sometimes even within the same document. "Turkey" and "Ottoman Empire" were the terms more commonly used.
London Protocol (1830) and Treaty of Constantinople (1832)
In the original French version of the London Protocol that established Greece as an independent kingdom, "l’Empire Ottoman", "Porte Ottomane" and "Porte" are used.
In the English version of the Treaty of Constantinople that marked the end of the Greek War of Independence, "Turkey", "Turkish", "Ottoman Sublime Porte", "the Sublime Porte" and "Ottomans" are used.
Pact of Halepa (1878)
I couldn't locate the original text of the Pact of Halepa, but I've found two mentions in near-contemporary newspapers that show a variety of names was used. The first one, from the July 16, 1896 issue of The Mercury (Australia), uses "Porte" when referring to the Ottoman government, and "Turks", "Moslem members", "Mussulman members" and "Mahometans" for the Ottoman people.
The second mention is from the May 29, 1903 issue of the Star, the evening edition of the Lyttelton Times (New Zealand). It uses "Turkey", "Turkish" and "Mohammedans".
Treaty of London (1913) and Athens Peace Convention (1913)
The Treaty of London and the peace treaty signed in the Athens convention concluded the First Balkan War. Both documents use "Ottoman Empire".
In a note the Great Powers send to Greece on February 13, 1914 concerning violations of the Treaty of London, "Turquie" is used instead.
Treaties of Sèvres (1920) and Lausanne (1923)
In the English version of the Treaty of Sèvres the Ottoman Empire is referred to as "Turkey", and the Ottoman government as "Imperial Ottoman Government" (preamble) and as "Turkish Government" (article 250). I couldn't locate the French (primary) or the Italian version of the treaty, nevertheless the English one is also an official one.
The Treaty of Lausanne was signed after the dissolution of the Ottoman Empire, however it's perhaps worth noting that there are various instances of "Ottoman" in it.
- Protocole (No 1) tenu à Londres le 3 Février 1830, relatif à l’indépendance de la Grèce.
- Arrangement between Great Britain, France, Russia, and Turkey, for the Definitive Settlement of the Continental Limits of Greece. Signed at Constantinople, 21st July, 1832.
- The Mercury, Thursday 16 July 1896.
- Star, Issue 7718, 29 May 1903, Page 2
- Convention de Paix d’ Athènes
- Peace Treaty between Greece, Bulgaria, Serbia, Montenegro and the Ottoman Empire
- Note des représentants d’ Allemagne, d’ Autriche-Hongrie, de Grande-Bretagne, d’ Italie et de Russie au Gouvernement Grec, en date du 31 Janvier/13 Février 1914.
- Peace Treaty of Sèvres
- Treaty of Lausanne