References

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Their paragraph is shown below, and the thing to look for is the idea of the "thing" just being a point and that the origin is coming from the coordinate system.

"Now, let V be the position vector extending from the origin of K to a particular point P, and V* be the position vector extending from the origin of K* to that same point. Assume that the origins of K and K* do not coincide; then V ≠ V*. The position vector is very definitely coordinate dependent and is not a tensor because it does not satisfy the condition of coordinate independence."

We take the opinion that the something isn't a vector unless it is defined by two  points which exist independent of a coordinate system.  (We allow someone to use a coordinate system to draw those two points that become a directed line segment--they must then prove it exists independent of coordinate system by removing their coordinate system.)