It's an integral part of every encoded video file. If you export a post-processed video into some ordinary *.mp4 file, it's always there.
But please explain a bit more what you regard as color banding and at what part of the pictures you posted you found that color banding. How you did see it. I didn't find anything. Nothing that can be called "bad" or "noticeable color banding" at all.
I downloaded the pictures and loaded them into an image processing software and tried to find differences. I wasn't able to find any visible difference. Only when I examined and compared single pixels I was able to find a very faint differences in the picture you called "screenshot" between the picture you called "recording". Between the other picture pair you called x264 and h264 I wasn't able to find any difference in quality. Absolutely none.
I used the difference operator between the two picture pairs, and the only differences that appear are due to movement of the spiked stars (I assume they rotate) and the spaceship's position is moved by one or two pixels as well. This is due to the fact the pictures are not made from the same point in time. They are at least one frame away from each other.
The diffs look like this:
Here, the spaceship moved 2 pixels horizontally and about half a pixel up between the two frames, otherwise the spaceship diff would be completely nonexistant. You clearly see the 2 pixels movement by looking at the ropes of the opened door at the bottom. This simply an object movement, no color banding.
Here, the spaceship moved 2 pixel horizontally and about 1 pixel up between the two frames. No color banding..
The spiked stars cannot be included in any comparison, because they rotated between the two frames. The background is completely black, which means no differences with the background textures.
In my opinion, every picture is as good as you can get for postprocessing. Really. There isn't anything to complain about, Not a single thing.