"small-scale, petty" 1853, American English, colloquial, in reference to towns; see one + horse (n.). Probably from earlier use in reference to a carriage, sleigh, plow, etc., "drawn by a single horse" (1750); also "possessing only one horse" (of a farmer); hence "petty, on a small scale, of limited capacity or resources; inferior."
Shortly afterwards I took a stroll over the town. It was what is generally denominated a "one horse town," and I would think a pretty small pony at that. Two stores, one grocery, a stable, and four dwellings made up the sum of its buildings. ["Daguerreotyping in the Back Woods," in Yankee Notions, March, 1855]
"insult, snub," 1962, from verbal phrase put down "to snub," attested from c. 1400 in this sense, earlier (c. 1300) "to lower, let down," also (mid-14c.) "to throw down, reject;" see put (v.) + down (adv.). To put (something) down "end by force or authority" (a rebellion, etc.) is from mid-14c. Compare set-down "a rebuff, a scolding" (1780).