Loosely speaking there are three different categories of vehicle dynamics control on your truck - it's all one system of course but it has three different jobs.
First is stability control, that acts on throttle and brake to try to keep you from spinning out of control or rolling over. If you were driving at highway speeds and suddenly swerve to avoid a deer or something it's going to reduce throttle and selectively apply individual brakes to try to keep your truck pointed in the direction you're steering, not spin around or go sideways.
Pressing the button on your dash momentarily does not disable that, it's always active (it turns off in 4L on 4wd trucks)
Second is traction control. That acts on the throttle when it identifies that one or both of your rear wheels are spinning faster than the rest of your wheels, or when it identifies that your vehicle is oversteering while on-throttle (the rear of the vehicle is swinging out from a loss of traction on the rear wheels while cornering). It will reduce throttle input under those conditions to help you launch from a stop on ice, and help avoid fishtailing or getting sideways.
That is what gets disabled when you press the traction control button on your dashboard. It allows you to spin your tires and to get the vehicle sideways at low speeds which some people do for fun but also can be useful if you're stuck in mud or snow, to clear the tread of your tires if mud is packed in or to sustain momentum if you're barely moving.
The last mode is called brake lock differential which works on non-LSD trucks to simulate a LSD using the brakes. Since your truck is 2wd that would just be on the rear axle, it identifies when one wheel is spinning a lot faster than the other under throttle and clamps the brake down on that wheel to send power over to the other wheel. When you have LSD as well, it enhances the operation of the LSD and helps it last longer because it doesn't have to work as hard when it's engaged and it doesn't need to be preloaded like they used to be. This is always active and is not disabled when you press the button on your dash.