The term "Ottoman" (Osmanlı) was a term which originally referred to the companions and followers of Osman, the first ruler of the Ottoman state. As the Ottoman Empire was established it came to be used to refer to the empire's ruling elite - those people who directly served the Ottoman sultan as part of the state apparatus - governors, soldiers, religious officials, and so on. Being an Ottoman was thus a sign of being a person of high status, and it was indeed something to be proud of. In fact, the Ottomans did not refer to themselves as "Turks", which could be a term of disparagement. The Redhouse Turkish dictionary published in 1880 defined the word Türk as meaning "a rustic, clod-hopper". Thus educated Turkish-speakers generally referred to themselves as Rûmîs, meaning "Romans", a geographical term which referred to the former Byzantine Empire, and those with a connection to the Ottoman state were proud to refer to themselves as Osmanlı.

As for whether non-Turks strove to become Ottomans - in fact, most of the highest-ranking Ottomans were not Turks, at least during the middle stages of the empire's history. From the mid-fifteenth century through to the end of the seventeenth century at the earliest, most of the Ottoman elite came from the Balkans and the Caucasus, and were not Turkish-speakers by birth. In the eyes of Mustafa Âli, a sixteenth-century Ottoman intellectual, "Turkish ethnicity had nothing to do with the Ottoman state except in the origins of its founders."^1

Ideas about the nature of Ottomanness changed dramatically during the nineteenth and twentieth centuries, but I'll leave it to someone more knowledgeable about those periods to address that aspect of the question.

1. Cornell Fleischer, Bureaucrat and Intellectual in the Ottoman Empire: The Historian Mustafa Âli, 1541-1600. (Princeton: Princeton University Press, 1986) 256.