The first type of fighting was called a "Fair Fight" or sometimes "Queens Rules." It later became known as Kid-glove Boxing because of the soft goat skin gloves they wore.
The second was Wrestling, but it was pronounced "Wrasslin or "Russlin." Actually there were two types of wrestling practiced in early America. I quote from one of my favorite books Albian's Seed, Four British Folkways in America, by David Hackett Fischer:
One was carefully regulated and elaborately staged in annual tournaments. The burly contestants commonly dressed in sleeveless vests, long tights tucked into stockings, and velvet trunks incongruously embroidered with delicate flowers.
It was a throwing game. If any part of your body besides your feet touched the ground you lost the bout.
Then there was another type of 'wrestling' which would begin with "bragging and boasting" and usually a bit of heavy drinking beforehand:
The...sport of bragging and fighting was also introduced to the American backcountry, where it came to be called "rough and tumble." ....it was a savage combat between two or more males (occasionally females), which sometimes left the contestants permanently blinded or maimed. A graphic description of "rough and tumble" came from the Irish traveler Thomas Ashe, who described a fight between a West Virginian and a Kentuckian. A crowd gathered and arranged itself into an impromptu ring. The contestants were asked if they wished to "fight fair" or "rough and tumble." When the chose "rough and tumble," a roar of approval rose from the multitude. the two men entered the ring, and a few ordinary blows were exchanged in a tentative manner. Then suddenly the Virginian "contracted his whole form, drew his arms to his face," and "pitched himself into the bosom of his opponent," sinking his sharpened fingernails into the Kentuckian's head. "The Virinian," we are told, "never lost his hold...fixing his claws in his hair and his thumbs on his eyes, [he] have them a start from the sockets. The sufferer roared aloud, but uttered no complaint." Even after the eyes were gouged out, the struggle continued. The Virginian fastened his teeth on the Kentuckian's nose and bit it in two pieces. Then he tore off the Kentuckian's ears. At last, the "Kentuckian, deprived or eyes, ears, and nose, gave in." The victor, himself maimed and bleeding, was "chaired round the grounds," to the cheers of the crowd.(p. 737)
Mixed Martial Arts..."is humanity itself compared with the Virginian mode of fighting," with its "biting, gouging and...you get the idea.