Is this all a product of contemporary historical nomenclature, did people at the time use it in reference to one another or did the states themselves label them as such?
The Ottomans did refer to themselves by their dynasty, though they didn't have one single name that they used - at least not in the Early Modern Period. While in modern times state and territory are thought of as a single entity ("France" means both "The government of France" and "The territory of France"), for the Ottomans this wasn't quite the case. So they commonly called their state Devlet-i Alîye-yi Osmânîye, "The Sublime Ottoman State", and they commonly called the territory which the Ottoman dynasty ruled over Memâlik-i Mahrûse, which is usually translated as "The Well-Guarded Domains".