I usually used the term “rider” which seems to be the most popular here in Australia and many other countries.
However, anyone in or on any vehicle or on an animal is riding. That doesn’t suggest they are in control.
In fact, the pillion could be a rider was well. (Or is that passenger?)
To indicate that the person is actually in control of the motorcycle, they surely have to be driving it.
So are they really a driver?
I see this a lot in mainstream media, particularly in the US.
Perhaps it is a misnomer, or maybe it is more correct than calling them the rider.
After all, the term “drive” can refer to urge or motivation, operating and controlling the direction and speed of a motor vehicle, travelling on wheels, and propelling or carrying by force in a specified direction.
Bikie and biker
Then there are the terms “biker” and “bikie”.
In the US, a member of an outlawed motorcycle club such as the notorious Hell’s Angels is referred to as a biker. In Australia, they are bikies.
But these terms are also misused by the general public to refer to anyone who rides/drives a motorcycle.
Some riders in Australia, particularly cruiser riders, can refer to themselves as bikers which would give the totally wrong impression to visiting Americans.
Meanwhile, Yanks think bikie sounds ridiculous and a little childish when referring to big, tough motorcycle club members.
Then there is the term “motorcyclist” which just seems twee, nerdy, scientific or technical.