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 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 “cab” originated from. Sometimes families would sell old carriages into hackney service, and they were called simply “hackney” or “hackney chaise”.
They made miniatures coaches and carriages for children. Sometimes they were replicas of the family carriage but without the folding heads or heads and with canvas for the upholstery.
Seats: 2 children
Horses: Generally, none. They used ponies, goats, even large dogs. For older children, they used a single horse.
Bath Chair & Invalid Carriage
Also known as an Invalid Chair
The 1800s version of a wheelchair (in the photo below), while a bath chair itself isn't a carriage, there was also a specialty carriage called an invalid carriage that was designed to accommodate an “invalid chair” or wheelchair. It would have been wheeled up a hinged ramped at the back. It was owner driven.
The most 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.