But why “weighted” instead of “heavy” or something that doesn’t mean “changed”?

Oh wait, is it because the tax is “weighted” based on the weight of the vehicle?

That is correct. We also have farm plates, they're red text on a white background. Those are meant to be used only during for the purpose of performing farm business. Towing, hauling stuff around, taking goods to the market (I see tons of them on the vehicles at the farmers market in Raleigh). I believe they are also subjected to a lower tax and (May?) not require their safety inspection.

Edit: the weighted plate is for commercial use and over 9000lbs total (vehicle included). This can also cover non-commercial if you're hitting that weight and not hauling a camper. There are carveouts, but the tax increases with the weight of the expected max tow. The state charges a road use fee instead of a state tax on registration.