Perhaps I should quote the full explanation of the term hooley-ann.

Hooley-ann:
"A roping term. The throw can be made either from the ground or on horseback. The roper carries the loop in his hand, and when the chance presents itself, he swings one quick whirl around in front of him toward the right, up over his head, and releases the loop and rope in the direction of the target. As it comes over, it is turned in a way to cause it to flatten out before it reaches the head of the animal to be roped. It lands straight down and so has a fair-sized opening.
It is a fast loop and is strictly a head catch, being especially used to catch horses in a corral. It is thrown with a rather small loop and has the additional virtue of landing with the honda sliding down the rope, taking up the slack as it goes." W. M. French, "Ropes and Roping, 1940, "Cattleman" vol. 26, no. 12 (May), pp. 17-30; quoted in Ramon F. Adams, 1946, "Western Words," p. 79.
"The rope has not been slung over the horse's head, for to sling it so would cause even the steadiest old horse to become excited. Using the hooley-ann, half a dozen men can rope mounts at the same time without exciting the horses." John M. Hendrix, 1935, "Roping," "Cattleman" vol. 22, No. 1, (June) pp. 16-17.

Houlihan has already been satisfactorily explained.

"Hooligan"- Sometimes mistakenly put in the song. Throwing one would be difficult, since this is a wagon used on short drives to carry fuel and water.

Honda- A knotted or spliced eyelet at the business end of a rope for making a loop. A Spanish term meaning 'eye.'

In Gargoyle's post- Cannon, the 'cowboy poet," seemingly has never thrown a hoolihan (or a hooley-ann).
Will James was an French-Canadian writer (Ernest Dufault) and illustrator of western stories (from Quebec) who had a little experience as a cowhand as a teenager.

Lighter's "Dictionary of American Slang" has errors resulting from poor selection of 'authorities' (such as Cannon and James). Overall a good work, but with flaws (this can be said of most references that try to "cover the waterfront"). For details, more specific references must be used.

"Thrown the hoolihand"- Reference for the additional 'd'? Certainly seldom heard.

The origin of the terms is unknown, although some speculate that Hoolihan comes from the Irish name. Neither term seems to have antecedents in Vaquero or Texas Spanish.
The definitions are lacking in Gilbert y Chavez, Vaquero/Cowboy Lingo, Chap. Nine. Lingo
Or click on to the beginning of this fascinating booklet (on line) Cowboys Vaqueros