A lot of texts and websites I see imply that there are essentially two definitions of a straight line on a flat surface. The first is the shortest distance between two points. The second is a path that is inertial.
Sometimes I see what I believe to be misinformation implying that both of these definitions will generate the same path not only on a flat surface, but a curved surface as well. Below is a picture where it seems clear to me that the globally shortest path is definitely NOT inertial.
A third definition could be a path that goes a tiny distance in the dx direction, and then a fixed proportion of that tiny distance in the dy direction, and then repeats the process. That also generates a straight line on a flat surface but its path also seems to be different when applied to a curved surface.
How many other definitions of a straight line on a flat surface could there possibly be? Is there some sort of group theory method that could demonstrate when we've run out of such definitons? And would any of those definitions generate the same path in general when they are applied to a curved surface? Or would they all generate a different path when applied to a curved surface?