"Reflection" is in common usage to denote mirror imagery, and that would lead one to think that there is only one concept of reflection for each dimension, I agree. But there is a somewhat extended notion more prevalent in mathematics that allows you to reflect over anything that you can project onto. Usually that means reflecting across a hyperplane (the case you considered) or reflecting across a line in a space of dimension greater than two, or reflecting across a plane in four-space etcetera. Using a bit of the terminology of linear algebra, we say that a mapping is a "projection" if doing it twice is no different from doing it once, so for every vector X, we have P(X) = P(P(X)). To reflect across the space we are projecting into, we want the projection P(X) to be half way between X and its reflection R(X) so P(X) = 1/2( X + R(X) ) and therefore R(X) = 2P(X) - X. In Chapter 8, we will be able to express this sort of mapping in terms of coordinates, or just by "adding mappings". It is convenient to introduce the "identity mapping" I which sends every vector to itself, so I(X) = X for all X. We can then express the defining property of a reflection as R(X) = 2P(X) - I(X) for all X, or R = 2P - I. It helps to have things in this form if you are going to express them to a computer.
Somewhat more technically, we express the projection of a vector into a line along a unit vector Z by using the dot product: P(X) = (X€Z)Z. Then R(X) = 2(X€Z)Z - X. More generally, if we have a plane determined by two perpendicular unit vectors Z and Y, then the projection is defined by adding the projections to the two lines, so P(X) = (X€Z)Z + (X€Y)Y and the reflection is defined accordingly. Of course when someone gives you something to project into or reflect across, they usually don't furnish you with an "orthonomal basis" like Y and Z, so part of linear algebra is showing how to extract that description from any of the many other ways that we can define a "subspace" to project into.
Does this make sense? There should be a diagram to accompany this of course, but perhaps you can make one for yourself? Or work one out to go with this comment?