The metal circle in inside of the tire is a "wheel," not a "rim." The "rim" is the outer edge of the wheel. I do not know why it aggravates me so to hear people call wheels "rims," but it does.
The lovely thing about English is that even if it may not be technical, everyone gets it
With bicycles it makes more sense to make the distinction, but that is because there is more of a structural separation between the spokes and the rim.
I used to call them rims, but I "figured it out". At the same time, I don't like the term "wheel" either, I think the noun is too vague for cars. Wheel is used for too many other things, like wagon wheels and cartwheels. It just doesn't feel right.