Max, consider a second traveler leaving a planet orbiting Betelgeuse who follows exactly the same acceleration profile as the one leaving earth that has been considered in this thread. They leave at the same time according to a rest frame common to earth and Betelgeuse. After achieving their final speed, they will be at rest in a different frame of reference.
What do you call the distance between them? Proper or coordinate? Variant or invariant? Do you use the same terminology that you use for the distance between earth and Betelgeuse or something different?