[gizmo686]

The "point on the number line" definition has always been non-rigourous. It is meant to imply the intuition that real numbers are what we typically think of as "numbers", notably that they extend to infinity, are ordered, and are dense (for any two distinct real numbers, there exists a real number between them). Of course from a rigorous perspective, this does not even suggest a difference between the reals and the rationals.

The line you are talking about in the rest of your post seems to be an 'unrelated' object that is used in geometry. I am not familiar with the formal definition of line that is used in geometry, but one way of defining a line is as the set of all points which satisfy "y=mx+b", for a given (m,b). A line segment would be the above definition with restrictions on the domain: x_0<x<x_f.