To find the possible configurations, I have written and run a Pascal computer program.
For each face, I represent the points by their distances to the three sides (these distance sum up to sqrt(3)/2).
In all possible ways, n points are chosen from a regular lattice of points on the tetraeder, with the restriction that we choose at least one vertex and use a good distribution of the points over the faces.
For larger n, I ran an alternative program, choosing many times at random n points that satisfy varying conditions.
The distance between two points in distinct faces is determined as the minimum of four geodesic distances: directly or via one or two other faces.
If the minimum of the n*(n-1)/2 distances is greater than the maximum up to then, the maximum is adapted, and printed with the accessory configuration.
The runs are confirming fairly well the results that I found and calculated on paper. The best configurations in the output are a bit worse than the best theoretical ones.