I think the term, "driving," as in, "I drive my car," comes from the horse-and-buggy days. "Car" comes from "carriage," and "driving" might come from "driving a team of horses," or "driving cattle."

But since horses are gone now, I don't think we "drive" our cars anymore, we ride in them. In that respect, "drive" better describes what I do on my bicycle, since I am the power behind it.

All this is only a guess, however.