So of course both of these are slight simplifications, but what is the connection between the two? If the earth is basically a circle, is an ellipse just a parabola stretched around a circle? Is a parabola just an approximation of a tiny part of an ellipse? How high do you have to be before you change your calculations of a trajectory?
The Math ain’t mathing.
Geometry!
https://en.m.wikipedia.org/wiki/Conic_section
They are similar and related but not quite the same thing. It does come down to whether it is orbiting or returning to a ‘flat’ plane, and while an eclipse isn’t quite the same thing as two parabolas put together, that isn’t too far off.