quarter horse vs saddle horse

quarter horse and saddle horse both are nouns.

Word	Noun	Adjective	Verb	Adverb
quarter horse	Yes	No	No	No
saddle horse	Yes	No	No	No

As nouns, saddle horse is a hypernym of quarter horse; that is, saddle horse is a word with a broader meaning than quarter horse:

quarter horse: a small powerful horse originally bred for sprinting in quarter-mile races in Virginia
saddle horse: a lightweight horse kept for riding only

Other hypernyms of quarter horse include mount, riding horse.

quarter horse (noun)	saddle horse (noun)
a small powerful horse originally bred for sprinting in quarter-mile races in Virginia	a lightweight horse kept for riding only