One could argue that a distinct American Way of War existed during the colonial period. From the earliest days, American commanders had to contend with Native American opponents who fought in a radically different style than European foes. This necessitated, in most colonies, building a serious militia system and a defensive network of forts and blockhouses quite unlike anything seen in Britain at the time. From these defenses, American colonists could coordinate and launch counter-raids in response to Indian attacks, or the occasional larger expedition aimed a destroying a Native American town.
Another trait that developed during the colonial period that mostly stayed true through the Revolution was that American soldiers who served on a longer-term basis than a militia callout tended to be fussy and demanding. Americans generally viewed longer-term military operations as undesirable, especially because the campaign season largely overlapped with planting and harvest times. As such, many Americans who joined provincial (or later, Continental) units were unattached, propertyless laborers (though the French and Indian War provided an exception to this). Fighting for a paycheck rather than a cause, American soldiers demanded pay equal or greater than what a common laborer would receive locally, even if that price was more than what British regulars would make; British colonial administrators organizing large armies for operations against Canada or the Caribbean had to either consent to pay more for (usually) worse troops or see provincial support evaporate. Likewise, Americans would refuse to work or fight if they did not receive the pay, clothing, food, and alcohol allotments in the quantity and on the schedule promised to them upon enlistment.
The Continental Army dealt with many of these same problems, but held itself together together thanks to a generally broad commitment to the cause and the ideals of the Revolution. Both the enlisted men and officers came to the brink of major mutinies at various points during the war, but were mostly returned to service by appeals to their patriotism and belief in the cause. Still, Washington and Congress had to balance what they wanted to do against what the Army would tolerate or consent to.
After the Revolution, much of the scholarship focuses on the growth and professionalization of the American officer corp. Military forces raised during the Washington administration - masterminded by Alexander Hamilton - tended to be expressly political formations. Some recruiting posters in the 1790s called on Federalists, specifically, to join, and there were several instances where groups of soldiers attacked Democratic-Republican newspaper presses and politicians who had been too vocal in their criticism of the administration. Elected president in 1800, Jefferson increased the size of the military and authorized the creation of a military academy at West Point to create recruit and train professional officers who would put duty ahead of politics.
After the 1820s, the officer corps came to pride itself on political neutrality (at least during service; several officers went on to run for or become President) and thanklessly doing hard, boring work. This included manning frontier outposts where army forces would, occasionally, be tasked with preventing new waves of colonists into the West from infringing on Native American territory (until new treaties were crafted that took more Native land "legally").
Sources and Further Reading:
Anderson, Fred. A People’s Army: Massachusetts Soldiers and Society in the Seven Years’ War (1984).
Anderson, Fred. The Crucible of War: The Seven Years’ War and the Fate of Empire in British North America, 1754-1766 (2000).
Brumwell, Stephen. Redcoats: The British Soldier and the War in the Americas, 1755-1763 (2002).
Chet, Guy. Conquering the American Wilderness: The Triumph of European Warfare in the Colonial Northeast (2003)
Eames, Steven. Rustic Warriors: Warfare and the Provincial Soldier of the New England Frontier, 1689-1748 (2011).
Grenier, John. The First Way of War: American War Making on the Frontier (2005)
Shea, William K. The Virginia Militia in the Seventeenth Century (1983).
Titus, James. The Old Dominion at War: Society, Politics, and Warfare in Late Colonial Virginia (1991).
Warren, Jason W. Connecticut Unscathed: Victory in the Great Narrangnsett War, 1675-1676 (2014).
Zelner, Kyle F. A Rabble in Arms: Massachusetts Towns and Militiamen during King Philip’s War
Martin, James Kirby, and Mark Edward Lender. A Respectable Army: The Military Origins of the Republic, 1763-1789 (1982).
Neimeyer, Charles P. America Goes to War: A Social History of the Continental Army
Ruddiman, John A. Becoming Men of Some Consequence: Youth and Military Service in the Revolutionary War (2014)
Rosswurm, Steven. Arms, Country, and Class: The Philadelphia Militia and the “Lower-Sort” during the American Revolution (1987).
Royster, Charles. A Revolutionary People at War: The Continental Army and the American Character 1775-1783 (1979).
Shy, John. A People Numerous and Armed: Reflections on the Military Struggle for American Independence (1976)
Ball, Durwood. Army Regulars on the Western Frontier, 1848-1861 (2001).
Crackel, Theodore J. Mr. Jefferson’s Army: Political and Social Reform of the Military Establishment, 1801-1809 (1987).
Cress, Lawrence Delbert. Citizens in Arms: The Army and the Militia in American Society to the War of 1812 (1982).
Skelton, William B. An American Profession of Arms: The Army Officer Corps, 1784-1861 (1992).