"Cab" also was used the way we use it today, to mean a vehicle with a driver for hire in a city.
For example, from the Manual of the Corporation of the City of New York, 1845: http://books.google.com/books?id=XloPAQAAMAAJ&pg=PA100&output=html
"Rates of fare of hackney coaches, carriages, or cabs..." The laws sound surprisingly familiar, regulating fares and requiring cabs to post the fare and their number. The usage of "cab" with that meaning seems to have been more common in British usage than American, but still shows up in American cities through the war. For example, in Incidents in the Life of a Slave Girlhere, others in New York use the terms carriage and hack, but an Irishman says "cab."