This is one of the ideas presented in this Chapter. (i.e. to identify the geometry with the invariant of the transformation). In 3D if x represents a reflection, it also represents the plane left invariant by that reflection. If x represents a rotation/translation it also represents the euclidean/infinite line left invariant by that rotation/translation. And if it represents a point reflection, it also represents the 1 point left invariant by that point reflection.

Framing it in terms of vectors/bivectors is what allows us to quantify these ideas. (i.e. have to write down your planes/lines/points with numbers at some point).

But you got the essence! The abstract/coordinate free/geometric/group theory ways of thinking about this are most insightful.

So in PGA, vectors are reflections, other isometries are combinations of reflections, and the geometry are the associated invariants. (including planes, lines, points and screw axi).

A fun extension is to replace the reflection by an inversion (reflection in a sphere). Vectors now become inversions (leaving spheres invariant), the rotors become the conformal group (composition of two inversions), leaving circles invariant, etc .. the corresponding Clifford Algebra is CGA (R(4,1), with a proper parametrisation).