A “cow puncher” (or “cowpuncher”) is sometimes called a “cowpoke” or the usual “cowboy.” The term “cow punch” was quite literal—the cowboys “punched” the cattle with sticks to bring them into the stockyards.
 
   
(Oxford English Dictionary)
cow-puncher (U.S.), a cow-driver in the western States
1878 S. L. CALDWELL Diary 24 Apr. in Colorado Mag. (1939) XVI. 152 At Hugo the *cow-punchers were assembling for the round-up. 1889 H. O’REILLY 50 Years on Trail 357 The town was full of cow-punchers, mule-whackers, etc.
 
(Dictionary of American Regional English)
cowpunch v SW
1882 Romspert W. Echo 189, Some of them have been cowpunching—as it is called—for many years ,and know every water for hundreds of miles around.
(...)
cowpuncher n
A cowboy. chiefly West Cr. cowpoke n, cow prod
1878 (Same as OED—ed.)
 
13 August 1874, New York Times, “Wyoming,” pg. 3:
The long horns or cow punchers is a general term to denote stockmen, and they, with the grange element, which is singularly well organized and includes all the ranchmen, now constitute the dominant party.
 
28 November 1879, Chicago Daily Tribune, pg. 4:
THE RED DEVILS IN UTAH.
(...)
A COW-PUNCHER.
 
27 October 1880, Christian Union, “Cattle Ranges in the Far West,” pg., 361:
Under him work the stockmen—cowboys, or cowpunchers, as everybody connected with cattle raising is called—from the different ranches, numbering often seventy or more men, and two hundred or more horses, for each cowboy has at least three, and often as man yas eight spare mounts with him on these occasions.