> In my opinion the circle should be positioned in an infinite plane, then random vectors should be drawn on the plane and ignored unless they intersect the circle.

After looking at the Wikipedia page, it seems that your method would be equivalent to the "random radius" aka "1/2" aka "ninja" method.

https://en.wikipedia.org/wiki/Bertrand_paradox_(probability)

This is a good demonstration of the paradox here. You phrase your method as "random vectors in an infinite plane", but even when it is equivalent to the "ninja" method, the ninja method doesn't seem right to you.