For the second example, sampling the radius uniformly leads to an error with a higher point density in the centre of sphere. The radius, r, has a weight proportional to r^2 or the surface area. So, one can take a uniform random sample, s between 0 and R^2, and take the square root of s. That will give higher probability density around larger radius. Then, the second picture should look like the first one.
Yes, didnt look at the details of the next methods. He adjusts for this in the next examples by taking the cube root of the uniform sample(not square root as in my post), which is the correct adjusting factor, as a small shell will have weight in proportion to difference between cubic powers, which is proportional to r^2.
