A horse-drawn vehicle is pulled by a horse, not by an engine. A horse-drawn cart, a horse-drawn carriage, a horse-drawn tram in an old city. You almost always put it in front of the thing being pulled. Though actually, it is not used for a person riding a horse — that is on horseback. Horse-drawn is about the wagon, the cart, the carriage. In most places today you meet it at festivals, on farms, or in old films, not in daily traffic, so the word often carries a feeling of the past.