A rodeo was work long before it was a show. The word named the job of gathering cattle, and only later the contests that grew out of it.
Etymonline traces it to Spanish rodeo, literally "a going round." It comes from the verb rodear, to go round or surround. Behind that is Latin rotare, "to go around," built on rota, "wheel." In Spanish the noun was also used for a pen for cattle at a fair or market.
Etymonline's first English record is from 1834, meaning a cattle round-up. Encircling a scattered herd is exactly the "going round" the Spanish word describes.
The sense most people know now came much later. Etymonline dates "rodeo" as a public show of horse-riding skill to 1913. By then towns had been staging cowboy contests for decades, but under other names.
Prescott, Arizona, is a good example. The Sharlot Hall Museum says the town added a "Cowboy Tournament" to its Fourth of July celebration in 1888. There was bronco riding, steer roping and cow horse racing, with admission charged to pay for prizes. Juan Leivas won the steer roping and tied for first in bronco riding. The committee later gave him a silver trophy as the tournament's "Best Cowboy." The museum notes the word "rodeo" was not used in Prescott until 1924. Earlier events were billed as tournaments, fiestas, cowboy contests or stampedes.
Prescott's claim to the oldest rodeo rests on paperwork. The museum's test has five parts: a planning committee, invited cowboys, admission, prizes and documented contests. Payson, Arizona, claims 1884, but the museum says those contests were informal and unrecorded. Pecos, Texas, has disputed the title too. Prescott registered "World's Oldest Rodeo" as a service mark in 1985.
Rodeo was not the only ranch word English took from Spanish. Wordorigins.org shows buckaroo is an alteration of vaquero, "cowboy," first found as "bakhara" in 1827. The rodeo's Latin root also turns up far from the ranch, in rotary.