Thanks for the Critchlow version.
The 'grazin' bit' in Thorp and Larkin is changed to 'Brazos bit' in his version. The 'grazin' bit' is a "small bit with a curb in the mouthpiece. It is a good all-round light-weight bit, does not punish a horse, and is used now in most states east of the Rockies" (Ramon F. Adams, "Western Words").
Critchlow's explanations are valid, except for a couple of errors. The Maguey (pronounced McGee in the song) is not a grass, but an agave, 'century plant', and is noted for the stiff, useful fibers in its leaves. The species A. sisalana, from Mexico, produces fibers made into rope and twine; it is not the only useful member of the plant genus. The family is known only from America and Australia.

'Dar la vuelta' is one of the many forms and mis-spellings of the Spanish term, probably already corrupted by the Vaqueros themselves; known to most cowboys as 'dally' or 'dolly'. The variant 'dolly welter' is used in a tale by Omar Barker about a tenderfoot roper who made a lucky catch and was advised from all sides "to take your dolly welter," but he retorted that he "didn't even know the gal" (Retold by Ramon F. Adams in "Western Words").   

In the article by Gilbert y Chavez (link in previous post, 'Vaqueros'), are illustrations of common saddle types including the vaquero saddle.