Speaking of Great Lakes . . .

I have family who live up in northern Ohio and we sometimes go up to Lake Erie and take a ferry to one of the little islands off the "coast" of Ohio. One of the posters inside the ferry said that Lake Erie has a current in a constant direction (East) technically making it a river. This seems a little odd but it makes sense because the water flows from Lake Superior through the others to the St. Lawrence River. If this is true then Lakes Huron and Ontario would be rivers as well, right? Or is this not a correct designation despite the constant current?