I have seen the Ottomans be named a variety of ways on old maps, and was wondering in what contexts these different names would be used, or what someone living at the time of its existence, living inside or outside of its borders, would refer to it as.
For the pre-modern period (before the 19th century) Turkey/Turkish Empire was purely European terminology. The Ottomans basically never called themselves Turks. Rather, they referred to the empire by a variety of other names, primarily Devlet-i 'Alîye-yi 'Osmânîye (The Sublime Ottoman State) for the state, and Memâlik-i Mahrûse (The Well-Guarded Domains) for the territory under Ottoman rule.
From the 19th Century onward the usage of these names likely began to change, though unfortunately I'm not as familiar with that period.