What does the intersection of the two graphs represent?

The graphs represent position vs time

When the graphs intersect, both students are at the same place and at the same time.  Their x and y coordinates are the same,

If two equations represent the paths of the two students, then when the lines intersect, both equations have the same solutions for x and y at that point.

whichever one it is, it's definitely not one that you included on the list you provided.