I was mystified by the expression "coordinate-free" when first encountering
GA , since the whole subject was quite obviously concerned with little else
than coordinates!
Eventually I realised that people generally have come to regard "coordinates"
as a singular term denoting an algebraic (vector) representation of some
geometric entity (point, line, plane, quadric, or whatever); whereas what a
Victorian mathematician called "coordinates" would now more be referred
to more specifically as "components".
It's the same kind of subtle shift in meaning that has overtaken the word
"data", originally the plural of "datum" denoting of a single reading of some
experimental quantity. Such evolution may well be significant, in reflecting
a greater level of abstraction in the way we think about these matters.
So the idea behind coordinate-freedom is rather to suppress reference
to individual components, which are ideally restricted to a standard
implementation layer, that --- once specified --- remains out of sight.
To some extent of course, this notion was already present even in
the usual Gibbs-Heaviside vector calculus: GA just carries it further.
Fred Lunnon