Berets of the United States Army
U.S. Army Special Forces wear the green beret because of their link to the British Commandos of World War II. The first Ranger unit, commonly known as Darby's Rangers, was formed in Northern Ireland during the summer of 1942. On completion of training at the Commando Training Depot at Achnacarry Castle in Scotland, those Rangers had the right to wear the British Commando green beret, but it was not part of the regulation uniform at the time and was disallowed by the U.S. Army. 
The 10th Special Forces Group (Airborne) had many veterans of World War II and Korea in its ranks when it was formed in 1952. Members of the 10th SFG began to unofficially wear a variety of berets while training, some favoring the red or maroon airborne beret, the black beret, or the green commando beret. In 1953, a beret whose design was based on that of the Canadian Army pattern, and which was rifle-green in color, was chosen for wear by Special Forces units.
Delta Force
The origin of the term operator in American special operations comes from the U.S. Army Special Forces (aka the Green Berets). The Army Special Forces was established in 1952, ten years before the Navy SEALs and 25 years before DELTA. Every other modern U.S. special operations unit in the Army, Navy, Air Force and Marines was established after 1977. In "Veritas: Journal of Army Special Operations History", Charles H. Briscoe states that the Army "Special Forces did not misappropriate the appellation. Unbeknownst to most members of the Army Special Operations Force (ARSOF) community, that moniker was adopted by the Special Forces in the mid-1950s." He goes on to state that all qualified enlisted and officers in Special Forces had to "voluntary subscribe to the provisions of the 'Code of the Special Forces Operator' and pledge themselves to its tenets by witnessed signature." This pre-dates every other special operations unit that currently uses the term/title operator.