Your challenge is to come up with such an experiment that can tell us that one person is absolutely correct, and another is wrong in their claim that they are "truly" stationary.
That's not even relevant, Max. As Newton has proved, you don't have to know a damn thing about "absolute" motion in order to do physics.
YOUR example of "Galilean relativity" was comprised of JUST TWO objects. YOUR claim was that neither could know, AS BETWEEN THE TWO OF THEM, which one was moving.
Yet you are ALSO the one to say this, aren't you?:
I just want to point out that the basic point that someone in a spaceship travelling at near C will experience less time going to the nearest star than someone on Earth watching is not theoretical.
That's correct. And you KNOW it is the spaceship moving. That's why it experiences less time PER THE DICTATES OF THE LORENTZ TRANSFORMATION. That's also what the "twin paradox" tells you.
The GPS tells us that, relative to the earth, the satellites, not the earth, are moving. It does that without
purporting to know, and without having to know, the "absolute speed" of either.
Using the CMB as a "cosmic rest frame," we now know that the entire solar system (indeed the entire galaxy) is moving at the rate of about 2 million mph toward the "great attractor." That doesn't affect the relationship between the earth and the satellites one bit, though.
For the last time, the question of ABSOLUTE MOTION is totally irrelevant to the issues we've been discussing. Those are questions of relative motion, not absolute motion.
THIS is false:
In order to truly prove that you are right you need to propose an experiment that one person can use to prove that they are truly stationary.
Being "truly stationary," or not, is irrelevant to question of who is moving relative to the other.
Get it yet? (I'm sure the answer is "no," but I pose the rhetorical question to you anyway).