I would not say the fringed top makes it more formal. The formality of the vehicle is more or less dictated by its original intended use. A Runabout is still a family, get-to-town vehicle, even if it has a top.
Without the top, Keely's vehicle is a Runabout. With the top, I guess I don't know what to call it, because it is not a folding top. Probably End-Spring Buggy is probably the closest term.]
Actually the term "runabout" means "a buggy without a top". They tended to be gentlemens vehicles for running errands in town or back and forth to work as they have a single seat. From what I can see in my "American Sleighs, Carriages, Sulkies and Carts", edited by Don H Berkebile, her vehicle closely resembles a Jenny Lind or possibly a single seat spring wagon with surrey top but I can't see the suspension well enough in her photos to say. Spring wagons most often had 2 or more seats and were family wagons. With a single seat they were business wagons for gentlemen. It is of course just a replica of some sort and modified to suit smaller equines so we would never be able to say exactly what vehicle it is
You could perhaps ask the builder what he intended it to represent.
In any case, as Myrna points out, none of these are "formal" vehicles.