I obtained these descriptions from the notes in the back of the Oxford edition that is linked for the group read. I recall seeing illustrations of the various carriage types over the years at Pemberley, but for reasons I could never quite figure out, I cannot search the archives using the several eons time frame. You may want to try that option too.

Barouche - a four-wheeled carriage, having a seat in front for the driver, and seats inside for two couples to sit facing each other.

Hansom - a two-wheeled carriage, holding two persons, with the driver mounted on an elevated seat behind.

Brougham - a one-horse closed carriage.

Phaeton - a four-wheeled open carriage with an occasional hood.

Landau - a four-wheeled carriage, the top of which may be open or closed.

Sociable - an open carriage with side-facing seats.

Victoria - a light, low four-wheeled carriage, with a collapsible hood and seats for two persons, with an elevated seat in front for the driver.