Early bicycle without pedals, you sat on it and pushed along the ground in strides with your legs.

Merriam-Webster

A lightweight wheeled vehicle propelled by the rider.

Wikipedia

A Hansom cab is a kind of horse-drawn carriage first designed and patented in 1834 by Joseph Hansom, an architect from Hinckley, Leicestershire, England. Originally known as the Hansom Safety Cab, its purpose was to combine speed with safety, with a low center of gravity that was essential for safe cornering.

Merriam-Webster

A light 2-wheeled covered carriage with the driver’s seat elevated behind — called also hansom cab.

Wikipedia

Beginning in the 17th century, they were originally horse-drawn carriages, later modernized as hansom cabs (1834), that operated as vehicles for hire.

Merriam-Webster

A coach kept for hire; especially : a four-wheeled carriage drawn by two horses and having seats for six persons.

A hack can also be used to describe a horse:

Merriam-Webster

A horse let out for common hire (2) : a horse used in all kinds of work b : a horse worn out in service : JADE c : a light easy saddle horse; especially : a three-gaited saddle horse d : a ride on a horse

Wikipedia

A cabriolet was a light, two-wheeled horse-drawn carriage with a folding ‘calash’ top, seating two persons behind the driver’s box. Developed in France in the early 19th century, the vehicle quickly replaced the heavier hackney carriage as the vehicle for hire of choice in Paris and London. The modern term (taxi)cab comes from “cabriolet” via “Hansom cab”.

Merriam-Webster

A light 2-wheeled one-horse carriage with a folding leather hood, a large apron, and upward-curving shafts.