You seem to be missing the central issue. Look back at Aurino's original question
Let me try and make it easier for you. You know the definition of acceleration, right? a = dv/dt
My question is, why is it that you can't possibly run any experiment to prove that the definition of acceleration does not properly describe the motion of bodies in space?
To which you replied by drifting off into reference frames specifications. Aurino's response brought the *definition* to a specific relationship between position, time and a standard mathematical process (a derivative)
Just so that [...] becomes more apparent, a better way to define acceleration is a = d(dx/dt)/dt. I would use the proper double-derivative symbol if I had it, but I'm sure you're knowledgeable enough to understand the equation without it.
The central issue here is that he has defined acceleration in a specific way. He claimed "you could not possibly run any experiment to prove that the definition of acceleration does not describe the motion of bodies in space".
Clearly "motion of bodies in space" is intended to be a reference to position as a function of time (at this point, position and time are nothing but numeric specification). The definition he gives is that acceleration is *defined* to be the second derivative of position with respect to time which is totally equivalent to telling you that the position obtained by integrating twice (a mathematical relationship).
Let us suppose you could prove his statement false. In order to run an experiment to disprove that statement, you would have to accomplish a number of things. First and foremost, you would have to be able to specify x as a function of t (otherwise the derivative is undefined). All of your complaints went to this issue, essentially pointing out that you could not accomplish that facet of the problem (which essentially is a proof that you cannot prove the equation wrong in any of the situations you describe). If you can not calculate a value, how can you prove the result of the calculation is wrong?
The second step (presuming you can specify x as a function of t for some experiment you are going to perform) just exactly how do you prove the equation is wrong if the equation *defines* acceleration? Aurino did not say that the result would agree with some other definition you might dream up; he stated it as the definition! I think your problem is that you have some other concept of "acceleration" which is confusing your thoughts. In fact I think you have difficulty comprehending the rigor necessary to work with exact definition particularly if what you have in your head is askew of what is being talked about.
Communication is very strongly dependent on understanding the meaning intended by parties trying to communicate. No matter how sure you are that you "know" the meaning of any word, it is always an assumption to believe that others are attaching the same meaning you intended.
Have fun -- Dick