When a casket is carried by and drawn by a horse it's called a horse drawn hearse.
It's simply called a horse drawn hearse and you can still have a funeral with a horse drawn hearse if you look around as some funeral homes do still use horse drawn hearses when requested.
My grandfather wanted to be taken to his grave in a horse drawn hearse and so we found a funeral home to do so.
It's the way I want to go to my grave one day as well.
Also a horse drawn carriage is considered a vehicle.
A vehicle is something that is used to transport goods or transport people.
So yes in most places a horse drawn carriage is considered to be a vehicle but not a motorized vehicle.
When operating a horse drawn carriage you still have to follow traffic laws and stop at stop lights, stop signs etc.
You also have to Yield the right away to pedestrians and other vehicles.
You don't have to have a drivers license to operate a horse drawn carriage nor do you need insurance or registration.
But you still must follow the laws that vehicles do when you operate a horse drawn carriage on public roadways.