I have never heard the term “linearly accelerating” reference frame. I have heard the term “uniformly accelerating” reference frame instead.

We are justified in using the term “uniformly accelerating” in reference to the Rindler coordinates because the Rindler $dt$ is a timelike Killing vector field. It is not unique in that regard. However, all timelike Killing fields in Minkowski spacetime share the property that they only cover part of the spacetime and the proper acceleration changes with the distance to the horizon. That is not an artifact of the coordinates, but part of the symmetry of the spacetime itself

I have never heard the term “linearly accelerating” reference frame. I have heard the term “uniformly accelerating” reference frame instead.

We are justified in using the term “uniformly accelerating” in reference to the Rindler coordinates because the Rindler $dt$ is a timelike Killing vector field. It is not unique in that regard. However, all accelerating timelike Killing fields in Minkowski spacetime share the property that they only cover part of the spacetime and the proper acceleration changes with the distance to the horizon. That is not an artifact of the coordinates, but part of the symmetry of the spacetime itself