The focus-directrix definition of a conic says that a (non-circular) conic is a collection of points such that there exists a line $\ell$ called a directrix and a point $F$ called a focus outside the line such that conic is the locus of all points $P$ such that the ratio of the distance from $P$ to $F$ divided by the distance from $P$ to $\ell$ is constant. This constant is called the eccentricity of the conic and the distance from the focus to the directrix is called the focal parameter. I have two related questions about well-definedness:
Considering some conics have more than one focus-directrix pair, how do we know that the eccentricity and focal parameter are well-defined? That is, why do they have the same value for all focus-directrix pairs?
Now, one might point to the fact that we know that a parabola has one focus-directrix pair, and that ellipses and hyperbolas have two, but I am not sure how to prove that. I know how to prove their existence, but why does a parabola have at most one focus-directrix pair, and why is there at most two for ellipses and hyperbolas?
The first question is important to me because we refer to "the" eccentricity and "the" focal parameter while I have been unable to find a source that proves that they are well-defined. As for the second question, there are several concepts such as the center of an ellipse, linear eccentricity, the latus rectum, and more that are easier to define or make well-defined if it is clear exactly how many focus-directrix pairs exist and where they lie.
An answer or a reference to a source would be greatly appreciated.