Actually your third screenshot illustrates a particularly well-behaved 8-bit display system. If you open that screenshot in Photoshop and measure with the color picker, you get a perfectly regular series - 20/20/20 - 21/21/21 - 22/22/22 and so on all the way. Each step adds exactly one 8-bit value.
In other words, and as silk-m already pointed out, this is the banding you get with an 8-bit monitor/video card. This is the best case scenario. The reason you don't see banding in photographs is that they always contain just enough noise to break it up.
Worst case scenario is highly irregular banding with color banding on top because the three channels aren't in step.
Unfortunately you can't "try" 10-bit hardware unless you actually buy it. 10-bit capable monitors are generally expensive. You also need a 10-bit capable video card. Until recently that meant NVidia Quadro or AMD Radeon Pro. Recently, however, NVidia started supporting 10 bit for their standard GeForces as well, using the "studio" driver.
This is a very welcome development, because it should put pressure on all the vendors for a wider implementation of 10 bits. This is 2019 and we really shouldn't need to put up with 8 bit processing anywhere.