The British Americans began distinguishing themselves from their British counterparts long before the Revolutionary War began. They were divesting themselves of British thought and behavior in the 17th century.
What do we mean by the Revolution? The war? That was no part of the Revolution; it was only a cause and effect of it. The Revolution was in the minds of the people, and this was effected, from 1760 to 1775, in the course of fifteen years before a drop of blood was shed at Lexington. The records of thirteen legislatures, the pamphlets, newspapers in all the colonies, ought to be consulted during that period to ascertain the steps by which the public opinion was enlightened and informed concerning the authority of Parliament over the colonies.
~ John Adams
They were not suddenly Americans in 1775. They were not subject to the erosion of liberties after the Glorious Revolution that the British were. Enlightenment rationalism and separatist Puritanism were American movements in greater measure than they were European movements.
The Americans were a happy and prosperous people, and wanted to stay that way, resisting the intrusions of the Crown that began in the 1760s. They were resisting, essentially, a foreign invader.
They were seating their own legislatures, establishing their own republics, drafting their own constitutions. Some of them thought that the states might coexist with Britain as commonwealths, with allegiances to the king, but in 1776, even that was not to be.
They did not identify as Britons. They were Americans.