Without wanting to throw fuel on the fire, I should say that engineers distinguish whether or not vectors are associated with a point as a matter of course. If I recall the terminology correctly, one of the principal statics textbooks refers to vectors attached to a particular point as “rooted” vectors.
As an aside, the concept John D was describing (without naming it) was that of a cross section of a fiber bundle. A vector field is a cross section of a manifold’s tangent bundle.
One is free to dismiss it if you like, but when you are doing things like GR it really is a useful idea.