Word for the driver of a horse drawn carriage

A coachman is the driver of a horse-drawn carriage.

I think OP is just confused because you ride a horse, but drive a car to get yourself from A to B.

It's also true that at least some cowboys rode the cattle (mostly they drove the cattle, obviously).

But in the case of a horse-drawn carriage, I think the usage for stagecoach is evidence enough...

driver of the stagecoach (that's 4450 hits in Google Books)

Strictly speaking, he's driving/urging on the horses (but he's also driving/steering the carriage).

The Victorian statutes, Volume 1 uses the phrase

carriage driver

So it appears that term was in use in 1866.