I don't want to go too off topic, but the rim is actually the outer part of the wheel that the tire sits on. The whole thing (not with the tire), is the wheel.
This picture may explain it better:
This is a diagram of the parts of a wheel. Notice that the rim is the outermost part of the wheel. It is part of the wheel, not the entirety of it.
This picture may explain it better:
This is a diagram of the parts of a wheel. Notice that the rim is the outermost part of the wheel. It is part of the wheel, not the entirety of it.