... for the second solution, the point is defined in its polar coordinates (radius+angle) which are selected independently ...

I think this could be written better. For if a random polar coordinate is chosen, it is assumed that its radius and angle are chosen in such a way that points equally fills the inside of a circle (square root for radius and linear for angle). But these are exactly the "same" points as in the "third, Cartesian Coordinates solution". I think it must be said that both, angle and radius are chosen linearly from their domains. This is not, I repeat, what is considered under "choosing a random (equally spaced) polar point".

There indeed was a mix-up on the page. Thank you for pointing this out. The applets and the explanations had to appear in the reverse order.

For the second solution, the point is chosen via its Cartesian coordinates, both chosen uniformly, with points outside the circle discarded. The probability is 1/4.

For the third solution, the point is chosen via its polar coordinates, both chosen uniformly, as suggested by E. T. Jaynes. This means that the point is in fact selected from a rectangle

[0, 1]×[0, 2π]

The probability is 1/2.