If you want to get technical, the types of wagons usually used for cabs (in the US at least) were cabriolets and Rockaways (also called carryalls). Hansom is technically just a make of cabriolet. Rockaways were very common street vehicles.
Cabs were often Hansoms or Rockaway coupes like the one seen in this cartoon from the late 19th century
Here's the station wagon Tsu Dho Nimh mentioned. Looks like a type of Rockaway as well.
They weren't technically cabs, but omnibuses were also still available for hire even as late as 1912
The two general categories of wagons in pre-car days were self-driven and coachman-driven. Rockaways, Victorias, omibuses, stagecoaches, Broughams, cabriolets (where 'cab" comes from) and landaus were common coach-driven wagons, while traps, buckboards, buggies, runabouts, two-wheeled work carts, commercial vans & trucks, and Phaetons were common self-driven wagons.
These terms mean nothing to the average reader, so you probably wont ever need to go into the particulars, but just for your own general knowledge.
If you really want to delve into the wagon/automobile scene of that time, here's a collection of issues of "The Hub", which was a car/wagon periodical. These are from 1910. Google books has older ones too.
https://books.google.com/books?id=Z...SAhVBSyYKHWyNDLcQ6AEIODAG#v=onepage&q&f=false
Also look for old issues of "The Rider & Driver", another old driving magazine. I've seen issues floating around Google books. Interesting stuff with lots of pics.