Player rating for tournaments across all installments of Heroes.

Even so, my silly example shows that someone who participated for 1000 days will get 10 points, and the factor from the first option basically doesn't matter at all, even though it's supposed to reflect the most important metric: how close their participation days are to the leader's! And the position they achieved as a result might be completely meaningless, but the factor from the second option significantly boosts their rating, and not entirely deservedly!

You get ten points like this:
(x+20)/2=10, where x is an infinitesimally small value based on the first (non-linear) calculation method. Am I understanding your point correctly? So, the whole thing is that after he gets 10 points, they will be "adjusted" to 20, according to my proposal. That is, all participants at the end of the long list will get 20 points, regardless of how they performed...

Note: We can abandon this idea. I can rework the second formula so that all points start from zero instead of 20... There are no problems with that. It's all feasible. I just consciously want to introduce a minimum threshold "for participation." That's the idea. If it doesn't work, then it doesn't work, and the formula will take on a strict form according to your comments, i.e., the last place in your exaggerated example will get (0.1+0)/2=0.05 rating points. And then they won't participate in tournaments anymore... :)