According to my Middle Eastern/African history professor, people in Arabia, Persia, East Africa and India knew the Osman Empire as simply Rome (just as how the Byzantine Empire was officially simply named The Roman Empire), because the change of religious and ethnic identity of the rulers wasn't as big a deal to them as it is from a Christian viewpoint. The main interaction with the region were distant trade routes into Europe through Constantinople, so really it makes no big difference whether a Greek Christian or Turkish Muslim is in charge since it's still the same trade routes trading the same wares through what is essentially just a continuation of the same state.

Somewhere as far as China, I doubt there was any real difference between the Roman, Byzantine and Osman Empires, since all of them were Rome anyway.

To put this in context, China itself has been conquered or split apart multiple times throughout its history, and been ruled by foreigners (such as Mongols and Manchus). To someone from East Asia, I believe there's a more substantial difference between the various ruling dynasties, variations of government and amounts of land controlled by each iteration, but from an eurocentric view of history it's just China.