A dish being "fired" means finish the cooking process, eg put into the fire.
When a meat comes out hot from the cooking process, it needs to "rest" a certain amount of time for the temperature to even out and the juices to be reabsorbed before being sliced/butchered. This is time that the diners now need to wait before they get their meal.
So, in David Chang's model, when the dish is ordered a prepared, dead chicken is "fired" and heated so that it finishes cooking. Then, while hot and still in the pot, presented to the diners. It's taken back, must rest outside of the hot boiling pot, and then is sliced into pieces and plated to be served and eaten by the guests. The remaining carcass is supposed to then be boiled down into a stock for a 3rd dish.
However, the resting and boiling steps could be skipped. Why does it all have to be the exact same chicken!? Have a presentation bird, another that is fired and rested (while the other is presented) you can carve and serve quicker, and make a large amount of bulk stock that can be served on order. The only difference is you don't have the exact same bird going through the whole 3 steps to the diners and instead have the 1. dead cooked presentation bird, 2. the carved bird, and 3. stock from a bird from yesterday.
Dead and boiled.
> What is "fired" in this context?
In cooking, that can mean applying direct heat in any number of ways. In this guess, I'm guessing that it means finishing (i.e. doing a last small amount of cooking for color and texture) the bird by roasting or searing.
> Is the presentation bird unrested? What does "rested" mean in this context?
In cooking, resting means to remove the item from heat and let it sit for a while. Food cooks from the outside in so the surface temperature can be much higher than the internal temperature. Resting gives some time for those to equalize. It lets the inside finish cooking using the residual heat on the surface.
> Why does the kitchen need to "butcher" the chicken that's "rested"?
It's a boiled whole chicken. After it's rested, it gets broken down into parts that the diner can eat. For chicken, that's usually breasts, thighs, wings, and legs/drumsticks.
> What does "slowing" the other, presumably presentation, chicken mean?
Resting and breaking down a chicken takes time, especially resting. By having a separate presentation chicken, it means the staff can be simultaneously resting some other cooked chicken and butchering it while the presenetation is being paraded around.