Given the size of the holes you are doing this might not seem like useful advice but I would bet that this is what the artist in the picture did, as those are not perfect circles. Outside of this you could use a small punch to get a perfect circle.
The way you get that to work is put a lot of paper (or some other thick material) underneath so that the "work" paper will be less likely to tear out.
For larger holes or if punches are not available
Whatever you need to do that feels comfortable so that you can do one continuous cut. As with all cutting make sure you are using tools that are sharp!
Should I only turn the knife on its place or really move it?
Keeping the work stationary for me is a problem especially for circles, as you are moving your wrist and arm around more, which could put strain (cramping from repetition) on you faster (which in turn is asking for small mistakes). This can also reduce your control of the cut. I am in favour of moving both the work and your hand at the same time. I find this makes the motions easier and they take less time.
Should I hold the knife vertically or diagonally on the paper?
While the cut would naturally be at a slight angle with what I am describing, I would try to avoid doing so and try to cut as vertically as possible. Cutting on a diagonal can weaken the edges of the cuts making more susceptible to damage.
It is definitely important to try and do the cut in one pass. Hesitation or multiple cuts can create nibs or stray fibers. This is fixable of course but you have to remove more material which can affect the shape. However if done right corrections like that can be hidden as most people won't notice.
Should I apply pressure or not?
Not sure of the best answer to this other than not too much. You want the knife to do the work. As long as your tool is sharp you should not have to put too much force into it.