5n2Mom,
Most of the terms we use are are reasonable well defined in our RV Glossary, accessible via a button on the menu bar near the top.
While there is no industry-standard definition, "Payload' generally includes everything carried in the truck except a 154 lb driver and a full tank of fuel. Pick-up bed or front seat makes no difference. For practical purposes it is the difference between the "curb weight" and the GVWR minus 154 lbs (the federally-defined standard weight for a human driver). If you carry extra passengers or the driver weighs 300 lbs instead of 154, the amount of gear and other cargo you can carry in the back goes down.
The TC has to be carried by the truck, so you need enough Payload for it, passengers and any gear you put into the TC, including water and propane.
A half ton or 150 or 1500 truck is generally built to a much lighter specification than a 250/2500 (3/4 ton) or 350/3500 (one ton), so there are some fundamental differences in strength regardless of the payload specs, which sometimes can be fairly close. Today, a half ton 1500 is likely to have a payload that a 3/4 ton would have been proud to carry 10-15 years ago, but that does not mean a whole lot. Just concentrate on the weights you need to carry and leave yourself plenty of extra margin, both for safety and for future growth.
I strongly recommend a 3/4 ton (2500) as a minimum for any TC. You will never regret having the stronger, more rigid chassis and typically greater payload that the larger, heavy duty truck offers.