Pony
A drawing of a pony
Pony was a term used specifically for small breeds of horses.
Christopher Pike owned a pony called Sir-Neighs-a-Lot during his younghood. He broke his leg during a rainstorm and Pike's parents were forced to euthanize him, after which Pike cried for a week. (SNW: "A Quality of Mercy")
In 2372, Molly O'Brien drew a picture of a pony. Her father asked to see the drawing and asked her if the pony had a name. She replied that she didn't know. (DS9: "Accession")