Ellipses have multiple definitions, how are these the same?

A link to the full video is at the bottom of the screen.
Or, for reference: https://youtu.be/pQa_tWZmlGs

The full video this comes from proves why slicing a cone gives the same shape as the two-thumbtacks-and-string construction, which is beautiful.

Editing from long-form to short by Dawid Kołodziej
...
https://www.youtube.com/watch?v=-3jPW_TGATI