It is not a rope trick. It is a rope throw. Check out Jim Bob Tinsley's "He was a'singin' this song". The song goes back to Charlie Willis, a black cowhand working for the Stadler Bros ranch in Texas shortly after the Civil War. He was a wrangler for the outfit and made some eight or ten trips from Texas to another ranch the Stadler Bros owned in Montana. It is common on the range for the wrangler to "catch out" whichever horse will be used by a cowboy on a given day. The cowboy will call out the name of the horse he is planning on riding that day and the wrangler will rope him out. The wrangler's job is to "Throw the hoolihan". One of the interesting verses given by Jesse Morris, as recorded by Lomax in 1950 (he was an 70 year old Texas fiddle player who knew Charlie when Jesse was a young boy and Charlie worked for Jesse's father) explains leaving Cheyenne, off to Montana, which seems a short trip.
Old Paint had a colt, down on the Rio Grand
The colt couldn't pace, they called her Cheyenne
Goodbye, Old Paint, we're leaving Cheyenne
Old time cowboys didn't like to start breaking colts until they were 4 or 5 years old. They wern't thought strong enough to carry a person til then. If Cheyenne was only 1 or 2 years old she would have been left at the home ranch while the herd was trailed North.
It is well know that most Anglo cowboy terms for gear and methods came from vacqueros of Mexico. I wonder if the original rope technique came from someone named Julian so the lariat (org. la riata, the rope) was thrown like Julian does it?