Same page says
PICKUP WITH CAB / CREW CAB WITH BOX-TYPE BED -- YES
Discussion: The utility or box bed has to enter weigh stations.
RENTAL TRUCK -- YES
Discussion: Rental trucks can be a U-Haul, Ryder, Sears, Budget, Enterprise, etc. A rental truck is a motor truck, according to CVC Section 410, and must stop at the weigh stations. Most scale facilities make this very clear with signs reminding drivers: "All Daily Rental/Moving Trucks Must Stop At Scales When Open." However, if the rental truck is a pick up, it does not necessarily have to stop (see "Pick Up " section above). If the rental truck is a flat bed or utitility bed, then YES, it does have to stop.
I don't see where it says unless owner says it is for private use. You register it commercial, you may need a lawyer when you go to court. In the case of rental trucks, they are almost always used to move personal belongings, and not for hire.