Also after watching the video from the StyleGAN2 team https://drive.google.com/file/d/1f_gbKW6FUUHKkUxciJ_lQx29mCq... now I know that original StyleGAN, images from which are apparently used for this "game", produces faces with a "water droplet" and phase artifacts, so I was able to spot few fakes just by looking for those things.

Only whichfaceisreal.com uses the original StyleGAN. The github.io links use StyleGAN2.

That's speculative at best. What's inorganic about the background in https://veedrac.github.io/stylegan2-real-or-fake/Curated/ffh... ?

How is it speculative? The background does not always allow for the discrimination to be made but in the random sample of ~20 faces I looked it was the main factor in maybe 1/4 of the cases (I was %100 accurate on this random sample). Of course, my random sample is not your random sample. We could probably do a controlled experiment to get at these kinds of attributions systematically.

I might also say that a second major discriminating factor is skin texture, especially at boundaries.

When not the ears, it's the background.

If the real pictures had the background removed, I'd have a very hard time scoring 100%.

That's the point of the hard version.