What does "Parameterizations are not unique" mean here?

Question

I am reading Betancourt's introduction on HMC. In the first chapter, I see the following statement

I am not quite sure what "Parameterizations are not unique" means. My hypothesis is that we could express the expectation in different ways, for example, as the very simple case below:
$$
f(q) = q;quad
pi(q) = q, qin [0,sqrt{2}]
$$
Therefore, the expectation for $f(q)$ could be calculated as $E(f(q)) = int_{0}^{sqrt{2}}q^2dq$.
On the other hand, we could let $q^prime = q-1$, then $q^primein[-1, sqrt{2}-1]$ and we can get a new integral for the same expectation value, i.e. $E(f(q^prime)) = int_{-1}^{sqrt{2}-1} (q^prime + 1)^2dq^prime$. I don't know whether this could serve as an example of "Parameterizations are not unique" or I misunderstood this sentence. Thank you.
P.S. Here is the link of the introduction

Dhanvi Sreenivasan · Answer

Yes, that seems right. Essentially, a parameterization of a sample space is a way of representing it. Imagine you are trying to find the expected rainfall over a city. The sample space is each point in that area, and the function takes a point in the space and returns the probability that rain will fall there.
Your parameterization of this sample space could be cartesian (i.e x and y coordinates), polar (radial distance from centre, and angle) etc. What this is saying is that the parametrisation is not tied to the system, but is up to user choice. But the expectation of an event occurring shouldn't matter, as long as the parameterization is described properly

What does "Parameterizations are not unique" mean here?

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