Your question exposes the importance of defining notions in physics unambiguously and universally in terms of "How to measure?".
As Einstein put it explicitly (however, referring specificly only to the notion of "simultaneity", and unfortunately only as late as 1917):
"We thus require a definition of simultaneity such that this definition supplies us with the method by means of which, in the present case, he can decide by experiment whether or not both the lightning strokes occurred simultaneously. As long as this requirement is not satisfied, I allow myself to be deceived as a physicist (and of course the same applies if I am not a physicist), when I imagine that I am able to attach a meaning to the statement of simultaneity. (I would ask the reader not to proceed farther until he is fully convinced on this point.)"
[Translation of the German original retrieved from http://en.wikisource.org/wiki/Relativity:_The_Special_and_General_Theory/Part_I#Section_8_-_On_the_Idea_of_Time_in_Physics]
Likewise, obviously, the notion of "(joint membership in one) inertial frame", a.k.a. "(mutual) rest" is meaningful only as far as a method has been declared and understood for deciding case by given case whether and to whom this characterization applied.
Only once the corresponding method has been provided and agreed on we can think concretely of
1. applying it to available observational data, and thus
2. experimentally testing any (pre-existing) models or expectations about "who was at rest to whom, in which trial".
Any underlying methods ("How to measure whether ...") are of course not themselves subject to experimental testing.
This requirement of "definition through a method by which to decide" must certainly be fulfilled by all notions which are intended to "differentiate experimental settings":
for instance, surely we wouldn't require by definition that all distinct participants should be "at rest to each other";
nor that all distinct members of an inertial frame should have "equal pairwise distances from each other";
nor that all distinct oscillation periods of some oscillator under consideration should all have "equal duration"; etc.
In these cases we need to declare and understand "How to measure whether ...".
But it should also be obvious that the requirement of being "defined through a method" cannot be upheld for all notions whatsoever. There must be certain sufficient notions recognized (presumed, or, in turn, granted) by which to express any further "methods" in the first place; i.e. notions which are considered self-evident (a.k.a. "axiomatic") and which therefore guarantee that the methods to be expressed are indeed unambiguously and universally comprehensible and applicable.
On this, Einstein also left a clue, namely:
"All our well-substantiated space-time propositions amount to the determination of space-time coincidences [such as] encounters between two or more recognizable material points."
[Translation of the German original retrieved from http://en.wikisource.org/wiki/The_Foundation_of_the_Generalised_Theory_of_Relativity#ch.3.p.776].
Before drawing any further conclusions, the task at hand for physicists (and with relevance to non-physicists) is therefore to use these axiomatic notions in order to express (and agree upon) some method for experimentally deciding "(joint membership in one) inertial frame", a.k.a. "(mutual) rest", in the first place. (A sketch of my own attempt is documented in my answer to the question "What determines which frames are inertial frames?" (PSE/a/70646).)
Remark added in response to a comment:
Certainly, there have been numerous tests of both Special Relativity and General Relativity.
Hardly. The very idea of wanting to have "experimental tests" of Special Relativity and/or General Relativity is utterly mistaken. The Theory of Relativity, as proposed by Einstein and elucidated especially by the above quotes, is a system of axioms, definitions (to be expressed by means of the axiomatic terms), and resulting theorems, concerning how to measure geometric and kinematic relations (between identifiable participants, a.k.a. "material points");
in particular dealing with how to measure
whether and which given participants had been "at rest" to each other (a.k.a. joint members of one "inertial frame"),
geometric relations between participants who had been at rest to each other (namely: values distance ratios), and
kinematic relations between participants who had been at rest to each other, but (especially, foremost) who had belonged to distinct inertial frames (namely: "speed ratios $\beta$").
But RT does not involve any experimentally testable hypotheses or models that any particular "material points" under consideration, in any particular trial under consideration, had had any such specific geometric or kinematic relaions. RT merely deals with the methods of how to measure whether; but those are not themselves experimentally testable, instead they are necessary pre-condition for expressing any experimentally testable hypotheses or models dealing with geometric or kinematic relaions in the first place (such as any experimentally testable models, and especially the so-called "standard models", of cosmology, of astronomy or astrophysics, of material science and chemistry, of particle physics -- you name it).
Note that the whole of RT (axioms, definitions of measuring methods, etc.) has remained and continues to remain intact and useful and uncontroversial even if any particular experimentally testable hypothesis or model about geometric or kinematic relaions of given "material points" had been found false (experimentally falsified) by using those very measuring methods of RT.
