Questions of identity like this are always difficult to answer, but your second option (true heirs of British cultural inheritance) is closest to the mark I think. This answer will mostly be drawn from Eric Kaufmann's 2002 article 'Modern Formation, Ethnic Reformation: The Social Sources of the American Nation.'
A noticeable American identity developed in the 13 colonies during the 18th century. Kaufmann (2002) describes the increasing use of the term 'American' by the colonists to describe themselves after 1750, but also that they had a dual identity, conscious of themselves as British but distinct from those in the metropole. This dual identity was made definitely separate during and after the revolution, when a new political identity, separate from Britain, was created, with new national icons, like George Washington, and symbols, like the flag. This separate national identity was further heightened in the war of 1812 (see Kaufmann 2002 again), which produced a new symbol of American identity, 'the star-spangled banner'. So Americans after independence (or at least the dominant ideological strand) did not think of themselves as British in a political sense.
On the question of whether the Americans saw themselves as 'protestant Brits', Protestantism was very important to American identity, but you seem to be implying that this made America different from Britain. In fact, Protestantism was a crucial dimension of British identity too, often being described as one of the pillars of this identity (until the pate 20th century), together with liberty / parliamentary sovereignty, war, and empire (e.g. described by Colley 2003 in her introduction). It's true that the nature of protestant identity differed between Britain (with the established Church of England) and America (with establishment of a state religion being forbidden in the US constitution), but as America was solidly protestant, this difference was more a political than religious one.
Ethnically and culturally, early Americans did conceive of their identity as of British heritage, though as you say, somehow more pure than that of the British themselves. Culturally (and politically) this was about liberty, and linked to this was an ethnic conception, that harked back to the Anglo-Saxons as the true origin of these ideas of liberty, ideas that had then been quashed in subsequent centuries, most notably by the Normans. (These ideas about Anglo-Saxon liberty had originated in Britain itself during the political conflicts over royal governance of the previous centuries). Just as the Anglo-Saxons had migrated to Britain, the most adventurous and freedom loving of their descendants had migrated to America. Kaufmann reproduces the words of historian Reginald Horsmann:
"The various ingredients in the myth of Anglo-Saxon England, clearly delineated in a host of seventeenth and eighteenth-century works, now appear again in American protests: Josiah Quincy Jr., wrote of the popular nature of the Anglo-Saxon militia; Sam Adams stressed the old English freedoms defended in the Magna Carta; Benjamin Franklin stressed the freedom that the Anglo-Saxons enjoyed in emigrating to England; Charles Carroll depicted Saxon liberties torn away by William the Conqueror; Richard Bland argued that the English Constitution and Parliament stemmed from the Anglo-Saxon period....George Washington admired the pro-Saxon history of Catherine Macaulay and she visited him at Mount Vernon after the Revolution.54 "
Then there's Jefferson "who proclaimed to John Adams after drafting the constitution in 1776 that the Americans 'were 'the children of Israel in the wilderness, led by a cloud by day and a pillar of fire by night; and on the other side, Hengist and Horsa, the Saxon chiefs from whom we claim the honour of being descended, and whose political principles and form of government we have assumed.'"
So in conclusion, the early Americans did not exactly consider themselves literally ethnically or culturally British, but that their new identity was an offshoot or culmination of the British (or really English) past.
Americans saw themselves almost as more British than the British themselves. The American Revolution was had at its core an issue of traditional English rights. Right hat were hard fought and won in the English cival war, and the glorious revolution. The American Revolution was a continuation of England's evolution into a more free and democratic society.
It wasn't until 1812 that Americans and Canadians began seeing themselves as differant nations.