How to make rendering loop selections work in Movie Studio Platinum 13?
Oh, I found it myself and leaving the answer here in case someone else will get confused by this because the solution is not obvious.

I was setting the checkbox here:



but if I use "Advanced options", Movie Studio ignores this setting!

I have to select it deeper in a hidden dropdown in the "Advanced options" window like this:



And later when rendering the full video, it is important to remember to open that dropdwon again and turn off the region rendering.

Oh, Sony, this is so confusing - two things doing the same, but effective only under specific circumstances.