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