The first horse races I went to, at a county fair back in the 50s, did not have the jockeys sitting on the horses' backs, but rather in sulkies drawn behind the horses. Trotting races. And from the following lines in the Music Man, after comparing the corrupting influence of pool (as compared to wholesome billiards), Harold Hill sings in Ya Got Trouble,
And list'nin to some big out-a-town Jasper
Hearin' him tell about horse-race gamblin'.
Not a wholesome trottin' race, no!
But a race where they set down right on the horse!
Like to see some stuck-up jockey'boy
Sittin' on Dan Patch? Make your blood boil?
Well, I should say.
I gather there was once considerable feeling against horseback racing. Hence the need to distinguish it, terminologically.
(The song is here: https://www.youtube.com/watch?v=LI_Oe-jtgdI )