Of course there are differences, a horse has internal organs and legs, a car has an engine and wheels.
My point remains though, in both circumstances you have a pilot who is not in full control of their vehicle.
If you want to take the alcohol out of the equation, then just change the car for one that can accelerate, brake and or change direction without the driver being able to stop it.
I could be wrong, but I’m guessing the common factor in horse related deaths is motor vehicles, not the horse rider. Ie I doubt many pedestrians are being killed. As with cyclists it’s the motorists doing the killing. So the “diminished responsbility” the horse riders have is getting them killed, not causing them to kill. In the the drunk driver example the odds are significantly higher than they will kill other people.