A carriage with a hooded roof which folds into two sections is found on a landau. However, this is not necessarily a small carriage.
you are probably thinking of a buggy
A buggy, chariot, and horse-drawn vehicle are just a few other names.
Most carriages were drawn by either one, two, or four horses. The only 3 horse carriage I could find was an English coach.
wagonette or Chaises phantom
Fiacre
A Carriage
A carriage, a buggy, stagecoach. a cart.
The proper, traditional term is a "Coachman", which applies to the driver of any kind of horse drawn carriage.
Hansom is a horse drawn cab.
A Russian horse carriage with three horses harnessed abreast is called a Troika. The two outside horses are bent slightly to the outside and must canter to keep up with the center horse which is harnessed straight in and trots.
The horse drawn carriage was invented in Hungary in the Fifteenth Century. It was built to carry a princess to her wedding but this early design had no steering as such and had to be dragged sideways by its team of horses to turn a corner. Development was slow but eventually the front axle was mounted on a turntable to provide steering and the horse drawn wagon was a semi-practical, if not very comfortable vehicle. Suspension systems evolved slowly but by the early nineteenth century most of the problems had been solved and the improvements in road construction and maintenance made the horse drawn carriage a viable form of transport. It was invented by Lu Ban. The carriage is still in use.
surrey
Martingale
An aroba is an alternative name for an araba, a carriage used in Turkey and Asia Minor, drawn by horses or oxen.
The Black Mariah is the name referring to a horse drawn carriage dating back to the early 1900's. The wagon doubled as a hearse or an ambulance.
hansom cabs
Winfield Scott was old and ill and unable to go to into. If he had a horse it was hooked to a carriage. Thus, no name ever mentioned...........