Can a line in a projective plane have just two points?
No. Let $A, B, C, D$ be four points that axiom 3 guarantees. The line $AB$ must meet line $CD$ in a third point $E$ on each of those two lines.
It follows that every line must contain at least three points since all lines have the same number of points. To see that, suppose $L$ and $M$ are different lines. Pick some $P$ on neither. Then join $P$ to all the points on $L$. The intersections of those lines with $M$ establishes a bijection.
Almost a duplicate of Number of points on a line in a finite projective plane