I can't see how the numbers on the site are accurate because how much you can put in the truck will be impacted by the GVWR which in turn will be impacted by the motor/options you have, etc. My truck (GVWR 11,200) on the site you provided lists a payload of 4,050lbs while the recommended payload that Ford stuck on my door is 3,423lbs. My rear axle weight rating is 7,000, the tires are 3,640 each.
I'm with you though, the numbers just don't add up...
The payload listed on a web site is usually just approximate based on the style of truck, while the payload sticker is specific to that truck including all options as built, and likely includes the weight of a 150 pound occupant in each of its seats. I don't know if your truck has seating for 5 or 6, but it sounds pretty close.
My 1999 F350 CC weights 6800 pounds, and has a GVWR of 9700, for a total payload of 2900 pounds. If I take away 900 pounds for 6 passengers, I have 2000 pounds left, which is what the camper package loading data certification says it can carry.
it seats 6 front axle rating 4400...rear axle rating 6100.....GVWR 10K payload 3575....I put a fifth wheel hitch and rhino lining in the bed which weighs appox 150lb.....also put on runnning boards another 60-70 lbs.....tongue weight of my 5th wheel is 1135lbs........when all loaded an gassed I have around 1440 pay loading remaining