I would agree, runabout would be the correct name. Light, inexpensive when horses were driven.
Steel wheels then were the norm, got lots more miles on them than rubber.
PLEASE do not hitch this to a horse without having the wheels checked by a wheelwright for soundness. They can tell if wood is really solid and dependable. Have the shafts checked over as well, easy to get brittle or rotten under any covering on them, so the break with any stress.
Mark each wheel nut amd wheel to the place it came off, if you remove wheels for checking. Nuts are right handed on one side, left handed on the other side. Wheels will have worn the axles to fit ONLY that wheel, unless they have roller bearings inside. Axles will probably need greasing and new leather washers put on before taking buggy out for a ride.
Sorry to say that runabouts are pretty common in the USA. Locally you can get pretty nice, usable ones for $1200 or less. Decoration ones go very cheap, but wheels are bad, so they are hard to move. Any carriage will not last very long if left outside in the weather. They are made of many small, light pieces of wood that warp and swell when left wet. Buggies are an amazing design product, but not good as decorations. Most last outside less than a year before falling apart.
The second carriage is no where close to being a runabout. More expensive to begin with, has all the "desirable options" for fancy carriages with 5th wheel cut under for turning, groom seat behind, independent shafts, lamps, patent dash and fenders. Looks modern, not an antique. That will cost what several runabouts will bring, together! Lovely vehicle.
Make sure your runabout is safe to use FIRST. Old wood is tricky, can break easily. Parts are not hard to find to replace worn items so you can use it. Do enjoy it as a real carriage. There were MANY of these made, they are not rare. Just don't expect to make lots on resale unless you discover a rare maker's tag while cleaning it up. Most of these could have the folding tops added, they had holes and brackets ready to use on them. Folding tops were their "optional equipment" when purchased.