The variation in the music was pointed out by Dicho, 06 Jun 02 (and others), in thread 42786.
The variant music is from Margaret Larkin, 1931 Knopf, 1963 Oak), "The Cowboy's Lament" pp. 30-31; Glenn Ohrlin (see citation above); Lomax and Lomax, 1938 "Cowboy Songs and Other Frontier Ballads," version 3 of the "Cowboy's Lament" (copied from Larkin without acknowledgement).
The music was tweaked further by Christine Turner in "My Home's in Montana," for publication in Ginn's school songbook, and the four-line three-verse lyrics were written by her. Turner took only the first two lines from the Larkin model, since "shot in the bowels" was not suitable for a children's song book (although children appreciate gore).

Digressing a bit, good folk singers vary their rendition according to the emphasis they wish to give to various points in the story. If one looks through printed versions of "The Cowboy's Lament" and "The Streets..." other variant music is found; e. g., the music of Powder River Jack Lee ("Cowboy Songs," 1938, pp. 82-83) for that line fits neither 'cockles and mussels' nor the usual "Streets..." music. "Wrapped in his wet blanket and cold as the clay," with 'blank-' sung to the highest note on the line (ugh!).
There are many others depending on the effect desired. Cf. the effect in the first line- "As I walked out (high notes)- in the streets of Laredo" (sung to low notes) with that of "As I walked in the streets of Laredo (rising to 'in the' then gradually dropping to 'Laredo' (as sung by Jules Verne Allen, my preference).

Not important, but where did the line "With foot in the stirrup I'll gallop all day" come from? The Ginn textbook, or scout song book or other?
Ohrlin quotes the lyrics from the school songbook as "With foot in my stirrup, I gallop for aye." Has anyone got the Ginn songbook?
The pony changes from gray to bay in some renditions. No one seems to have come up yet with "my pony is gay."