During my research, I found a few random types of carriages that were used in the 1800s that don't fall into any neat category, generally because they were used for a very specific purpose.
These refer to any carriage that is available for hire. While many types were used, the most common type of carriage used was the cabriolet and were called hackney cabs, which is where the term “cab” came from. Sometimes families would sell old carriages into hackney service, and they were called simply a “hackney” or “hackney chaise”.
They made miniatures coaches and carriages for children to be pulled around the grounds. They came in any style you can imagine (though replicas of their family carriages were especially popular), but they didn't have heads and used canvas for the upholstery.
Seats: 2 children
Horses: Generally, none. They used ponies, goats, even large dogs. For older children with larger carriages, they might use a single horse.
Bath Chair & Invalid Carriage
Also known as an Invalid Chair
A bath chair is the 18th and 19th-century version of a wheelchair that was meant to be pushed by a person from behind. While this in and of itself is not a carriage, there were closely associated to invalid carriages. Some invalid carriages were vehicles in and of themselves that had a hinged ramp down the back that allowed for a bath chair to be wheeled onto it (kind of like a modern wheelchair accessible van). But most were basically bath chairs with a harness in the front for a horse. None of these were meant for long-distance travel and were only used around town.
The iconic style of sleigh, the cutter was the most popular sleigh in the 1800s. Instead of wheels, it had runners to allow it to be pulled across snow and ice.