Should half-hyperbola get more attention than hyperbola?

Author:
Scrooge
Topic:
Hyperbola
Making conics as Loci: set of points with the same distance from a circle and a point - these seem to be half hyperbolas and maybe ellipses, but can we prove it?
You can also get similar shapes if you get the set of points with the distances from two circles equal. Based on this kind of definition of hyperbola, would not it be better if the half-hyperbola got more attention than hyperbola itself? So hyperbola is a set of points where the differences of their distances from two distinct predefined points are constant (and not too large, to comply with the triangle-inequalities). But it matters how we compute the difference: we may take A-B or B-A, and each of the possibilities yield a half-hyperbola.