[ColinWright]

I suspect you are talking about formal formulations of abstract geometry. That's not what we're talking about here. Here we are talking about lines as being sets of points in the plane that satisfy an equation of the form ax+by=c. Solutions (x,y) of that are said to be a line, although they themselves are points.

You can deal instead with Euclid's axiomatization of geometry, and there "line" is an abstract thing defined by two points. Different animal, although seldom explained clearly by teachers, who often themselves don't really understand what's going on. (Although some do, and don't get the chance to explore these things because of the pressure of the curriculum, and students who don't care, but need to pass.)

All too often people get confused about this and are told to shut up by their teacher, whereas in fact the student has had an insight, and demonstrated deeper understanding.