Is a displacement a vector, a line segment, or something else? It probably seems ridiculously naive of me to ask such a basic question, but I have a need to use accurate language.  Typically, I think of a displacement as a directed line segment whose end-points are the locations of a particle at an initial time, and at a subsequent time.
Or, in the absence of such a particle, a displacement is a directed line segment between two points.
A displacement is not a vector.  A vector is defined either as a unique mathematical object in an abstract vector space, or as an equivalence class of parallel directed line segments of equal magnitude and sense, or some variant, thereof.
So, by my definition of displacement: Consider a particle initially at point $\mathscr{A}_0$, which moves along an arbitrary path to point $\mathscr{A}_1$, which is one meter distance from $\mathscr{A}_1$. A second particle is initially located at $\mathscr{B}_0$, which is not colocated with $\mathscr{A}_0$.  The second object moves along any other arbitrary path to point $\mathscr{B}_1$, one meter distance from $\mathscr{B}_0$, along a line parallel to the line determined by $\mathscr{A}_0$ and $\mathscr{A}_1$.  The displacements of these particles are equal in direction and magnitude, but are not identical displacements.
So, I ask, is my definition of displacement commonly accepted? 
For some of my  motivation for asking this question, see the introductory chapters in Gravitation by Charles W. Misner, Kip S. Thorne & John Archibald Wheeler or the introductory chapters in Modern Classical Physics Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics, by Kip S. Thorne & Roger D. Blandford.
 A: 
So, I ask, is my definition of displacement commonly accepted?

No. You're defining an alternative notion, which is not standard in the context of Newtonian mechanics. By the standard definition, your two displacements are the same displacement.
If you look at old physics textbooks, ca. 1930, you can find variations in these definitions, e.g., I believe in Millikan and Gale they define a vector as having a location. But nowadays the definitions are standardized so that displacements are vectors, and vectors are "portable."

For some of my motivation for asking this question, see the introductory chapters in Gravitation by Charles W. Misner, Kip S. Thorne & John Archibald Wheeler or the introductory chapters in Modern Classical Physics Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics, by Kip S. Thorne & Roger D. Blandford.

In the context of general relativity, neither positions nor finite displacements are vectors. Only infinitesimal displacements are vectors, and vectors that live at different points cannot be compared except by parallel transport. Also, "vector" in this context refers to a four-vector with the correct transformation properties for a four-vector.
