Player rating for tournaments across all installments of Heroes.

I don't understand what formula is used to calculate the third option?
The results from the first two options are added together for each player, and then divided by 2. Then the third option seems to fit.

Yes. It's the arithmetic mean.

A player from the first stage, who is 1000 times behind the leader of his stage, has the same PURE GAME RATING as a player from the fifth place in the second stage, who is only 8 times behind the leader of his stage (8*8=64 -> approximately 63). Is this correct?

You understand everything correctly. I commented on this point a little higher:

"the minimum number of points should be significant, because even the last place in the table should bring a tangible (moral) result... Let's say 50 people played in the tournament. 2/3 of them will receive a rating of 1. And those who didn't play will receive a rating of 0. Is that a big difference? And was it worth the effort?

My suggestion is to set the minimum rating to 20 (or 10)."

Based on this idea, both participants you are talking about will receive 20 points, even though one of them didn't earn even one. But he participated! And he was rewarded for that, with an eye toward the future...