One enquiry leads to another...

The only horse-drawn vehicles which I knew as a child in Ireland
were the Trap (with pony) and the Irish Jaunting-car.
Much later I learned of Fiacres,
named after the Hôtel de Saint-Fiacre in Paris,
where carriages gathered for hire,
and that Saint-Fiacre was one of three Irish saints called Fíachra, namely
St Fiacre of Breuil, (died 670), who built a hospice for travellers
at Saint-Fiacre, in the Seine-et-Marne.

So I wondered if the Hackney carriage came from the inner London district of that name.

Indeed not.
It comes from the French Haquenée, originally denoting a quiet, moderately-sized mare
suitable for ladies, and thus also for pulling carriages.
The horsey verb 'to hack' is from the same source.
Then I wondered about the Landau, the Post-chaise (or Shay)
the Brougham...the Barouche...

and found this interesting web-page,
on which no mention was made (before I e-mailed the author)
of the Berlin[e] (now used in French to describe a ‘sedan’ motor-vehicle
[‘saloon’ in British English],
which term comes from the rectangular configuration of a ‘sedan-chair’
[sedio in Italian]). 
Break (in French - Shooting-brake in English)
and Cabriolet, (now also abbreviated to cab in English, as in taxi-cab)

also denote the designs of modern automobiles.

A Dog-cart can be a horse-drawn vehicle designed to carry dogs as well as humans,
or one drawn by a large dog, especially in the Low Countries,
for delivering milk and eggs.

click to enlarge

A photochrom from the late 19th century showing two pedlars selling milk from a dogcart
near Brussels, Belgium.