Quote by JDoolin
Measurements, as a rule, are observer dependent, but any given measurement is observer independent, because the reference frame of the observation device is already determined.

You seem to be using a strange definition of "measurement" that requires observers to be intentionally naive. Why in the world would anyone try to measure the angle between AB and AC with a moving protractor?
All
geometric quantities are invariant under coordinate transformations. In Euclidean space, geometric quantities include angles and distances. An angle is always measured between two lines at the point they intersect. A distance is always measured between two points along the line that connects them. In Minkowski space, geometric quantities include angles, distances, and relative velocities. Relative velocity is really just the "angle" between two worldlines.
I've used the term "relative velocity", but you should note that ALL geometric quantities are
already "relative". An angle is always an angle between two lines. One cannot say "The angle of line AB is 30 degrees", that makes no sense. Likewise, a distance is always a distance between two points.
I don't know. All I can guarantee is that when you made the measurement, you referenced your measurement from some origin, and in order to visualize that distance, I must reference it from some origin.

You realize that your inability to answer this question unambiguously
proves that Dale is in fact not using a coordinate system?
Quote by JDoolin
But if you are giving a distance, you already have a continuous map between the two points.

You'll have to explain. A continuous map from what space into what space?
In a unit displacement vector, there is a continuous mapping of some onedimensional space from 0 to 1.

This statement makes no sense. You don't seem to be using the words "vector" and "mapping" correctly. Furthermore, I have not once made any mention of vectors, so it's irrelevant anyway.