Use the word hackney to mean either an old-fashioned type of horse-drawn carriage, the type of horse used to pull such carriages, or a taxi cab. If you enjoy carriage rides through the park, you'll want to hire a hackney.
The original Hackney is a British place name, an area that's now part of South London but was once rural land covered with horse pastures. The small horse known as a hackney was named after the location, and so were the carriages-for-hire such horses pulled. As automobiles replaced horses, motorized taxis borrowed the name hackney, or sometimes hackney carriage, both of which are still used today in the U.K.