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1650s, powny, from Scottish, apparently from obsolete French poulenet "little foal" (mid-15c.), diminutive of Old French poulain "foal," from Late Latin pullanus "young of an animal," from Latin pullus "young of a horse, fowl, etc." (see foal (n.)) [Skeat's suggestion, still accepted].
German, sensibly, indicates this animal by attaching a diminutive suffix to its word for "horse," which might yield Modern English *horslet. Modern French poney is a 19c. borrowing from English. Meaning "crib of a text as a cheating aid" (1827) and "small liquor glass" (1849) both are from notion of "smallness" (the former also "something one rides"). As the name of a popular dance, it dates from 1963. The U.S. Pony Express began 1860 (and operated about 18 months before being superseded by the transcontinental telegraph). The figurative one-trick pony is 1897, American English, in reference to circus acts.
1824, in pony up "to pay," of uncertain origin. OED says from pony (n.), but not exactly how. In other sources said to be from slang use of Latin legem pone to mean "money" (first recorded 16c.), because this was the title of the Psalm for March 25, a Quarter Day and the first payday of the year (the Psalm's first line is Legem pone michi domine viam iustificacionum "Teach me, O Lord, the ways of thy statutes").
[in all senses fr the thing being small like a pony; the student senses, which have or have had horse and trot as synonyms, may also suggest something that carries one, gives one a free ride]