That is not what was meant. Here is a better rephrasing:
Let X be a set of points not all of which are collinear. Then, there are two points a, b in X such that the line l passing through X only passes through a and b.
Let X be a set of points not all of which are collinear. Then, there are two points a, b in X such that the line l passing through X only passes through a and b.
I don't see how this rephrasing changes anything. Of course there are two points a and b because again, the definition of the problem leads naturally, obviously, and definitionally to this result.
Consider a 3x3 grid. It satisfies this argument.
Can you have a set of points Y on a plane such that Y satisfies the following? Given any line passing through 2 points on Y, there is also a third point in Y that it passes through.