For example, if I was hit by a bicycle, should I tell the police a "cyclist" or "bicyclist" hit me. If either one, which one is more common in AmE?
Strictly speaking, cyclist includes someone riding a cycle with a number of wheels other than two: unicycle, tricycle or quadricycle.
However, the use of bicyclist nowadays is not very common. Unless of course there was a mixed group of bicyclists and tricyclists and you wanted to be specific.