Why does this property of reparametrizations of surfaces matter? I'm reading A. Pressley's book "Elementary Differential Geometry" (2nd Edition) and at the end of section 4.2 they start talking about reparametrizations of surface patches (or charts) on smooth surfaces. For reference, if $S$ is a smooth surface and $\sigma : U\to S\cap W$ is a regular patch for $S$, and $\Phi : \tilde{U}\to U$ is a diffeomorphism, then $\tilde{\sigma} = \sigma\circ\Phi : \tilde{U}\to S\cap W$ is a reparametrization of $\sigma$, and $\tilde{\sigma}$ is a regular surface patch.
Then, later on in 4.2 they say

These observations [about reparametrizations or regular patches] give rise to a very important principle that we shall use throughout the book. The principle is that we can define a property of any smooth surface provided we can define it for any regular surface patch in such a way that it is unchanged when the patch is reparametrized.

My question is, why should this matter? What could we possibly gain out of arbitrarily reparametrizing our surface patches? When could a definition break under reparametrization?
To give an example of how they use this principle, they later (in section 4.3) define a smooth map between smooth surfaces $S_1$ and $S_2$ to be a map $f : S_1\to S_2$ such that, for any regular surface patches $\sigma_1$ on $S_1$ and $\sigma_2$ on $S_2$, the map $\sigma_2^{-1}\circ f\circ\sigma_1$ is smooth. They then proceed to check that under reparametrization ($\tilde{\sigma_1} = \sigma_1\circ\Phi_1$, and $\tilde{\sigma_2} = \sigma_2\circ\Phi_2$) that $\tilde{\sigma_2}^{-1}\circ f\circ \tilde{\sigma_1}$ is still smooth.
Why does this matter? Of course it works under reparametrization because we don't care what two patches $\sigma_1$ and $\sigma_2$ we choose in our definition, so it shouldn't matter if we reparametrize them. Why do they make such a big deal about this?
My first thought was that maybe this "principle" allows us to define something on an entire surface by only checking that it is true on an arbitrary fixed regular patch and on reparametrizations, but this doens't make sense because reparametrizations don't change the image of the patch, meaning if the patch doesn't cover the entire surface then we can't really talk about the property as being true on the entire surface because, well, we haven't checked it on the entire surface.
So what's the deal? Does this really matter as much as the book emphasizes that it does?
 A: The point is that you can define intrinsic, coordinate-invariant properties by (1) defining the property in a particular coordinate system, and (2) arguing that the property is invariant under reparametrization. This approach is often more "tangible" than developing more abstract coordinate-invariant machinery (such as high-order tensors). (Still, there is a kind of conservation of mathematical labor going on: the proof of (2) is doing the heavy lifting in this style of definition. We may not have to develop an abstract, coordinate-free apparatus, but we still have to prove invariance.)
A classic example is the down-to-earth definition of the index of a non-degenerate critical point. Given a non-degenerate critical point $p$ of a function $f$ (where the differential $df$ vanishes), define the index of $p$ to be the number of negative eigenvalues of the Hessian matrix computed in a particular coordinate patch. To show this definition is intrinsic, depending only on $p$ and $f$, we need to check the number of negative eigenvalues doesn't depend on the particular coordinate patch we chose. In other words, we need to verify (2). In this case, the verification is supplied by Sylvester's law of inertia.
