What is the most fair way to scale grades?

This answer proposes another definition of a "fair scale" and "fair exam" than the one proposed in the question.

We can approach this question from an information theoretic standpoint. In this perspective, grades should be informative about some underlying quality of students to solve certain problems. As such, there is a "true distribution" of various skills and abilities students have. Unfortunately, most likely these qualities are multidimensional but we need to "compress" these into an ordinal scale. This entails making some strange judgments such as "making a typo in an equation is 0.3 times as bad as accidentally multiplying both sides by zero". But suppose we have obtained some acceptable scale expressed as integer scores from 0 to 100. I am suspicious of cardinal scales and therefore I attach only ordinal meaning to these numbers (for now).

Importantly, if we would observe the results for the entire student pool there would be no need to ever rescale. The need to rescale arises because we observe different exams (information structures about the hypothetical score of students on "the true scale") for different parts of the student pool and want to make the scores between students comparable. In particular, if I believe to have set two similarly difficult exams to two random samples of 1000 students but in one case all students receive 0 points and in the other exam all students receive 100 points, then I should revise my belief about whether I truly set similarly difficult exams. If the sample size for each group is only 10 students, I won't update my belief as much and will be more reluctant to rescale the exam.

Now let's say for simplicity that we have observed the scores of two exams for the entire pool (or for each exam a sufficiently large sample) of students. Let's suppose that a density estimate of the distribution of scores of each exam is given by