Seems pretty clear to me -- If a point is on both sides of the line going through it, then there are trivial examples (as shown above) where no solution is possible. Indeed, no solution would be possible for any diagram with an odd number of reds or greens -- we have to pass a line through the diagram so that "at most half" of the points are on each side. Take the basic case of one point of each color. Pass the line through them. Counting them both as on both sides of the line, more than half are on each side.

Thus the puzzle clearly means to count a point on a line as not on the left of the line and not on the right.