I am a little bit confused about the definition of Sobolev spaces on manifolds. I organized my thoughts into 3 parts.
(1) A lemma in Spectral Theory (Chapter 6.2) by David Borthwick says that
Suppose that $F:\Omega \to \tilde{\Omega}$ is a ($C^\infty$ -)diffeomorphism. Then the pullback $F^*: u \mapsto u \circ F$ defines a continuous bijection $H^m(\Omega) \to H^m(\tilde{\Omega})$ for each $m\in\mathbb{N}$
In my mind, this lemma basically states that "being Sobolev" is preserved under coordinate changes. However, if we consider (as an example) the diffeomorphism between the interval $I = (-\frac{\pi}{2},\frac{\pi}{2})$ and real line $\mathbb{R}$, then clearly the constant function $1 \in H^2(I)$ but not in $H^2(\mathbb{R})$ (actually, not even in $L^2(\mathbb{R})$).
I believe we need to require the Jacobian of $F$ and its derivatives (of all orders) to be bounded?
(2) A classical way to set-up Sobolev spaces on compact manifolds $M$ is via partition of unity: Let $(U_k,\phi_k)$ be a finite cover of $M$ where $(U_k,\phi_k)$ are local coordinates and $\eta_k$ be a partition of unity subordinate to this finite cover. The norm for $f \in C^\infty(M)$ is defined by $$ \left\Vert f \right\Vert_{H^m(M)} = \sum \left\Vert(\eta_k f) \circ \phi_k^{-1}\right\Vert_{H^m( \phi_k(U_k) )} $$ The idea here is that although the norm depends on the partition of unity and cooridnate maps, the finiteness of norm does not, by the above lemma. However, as the example shown before, if the boundedness of Jacobian of transition maps and its derivatives is not required, then "being Sobolev" might not be preserved under coordinate changes.
I believe the partition of unity plays an important rule here, because it allows us to shrink the domain of integral to a (smaller) precompact set. Then by the smoothness of transition maps on a larger domain, the Jacobian and its derivatives are bounded on the smaller domain $\phi_k(\textit{supp}(\eta_k))$, right? Also, one can show that this definition is independent of the choice of finite cover and the partition of unity.
(Edit: This link Why is partition of unity required in definition of Sobolev space on manfolds? basically provided an affirmative answer to question (2).)
(3) In this post Definition of a section of a vector bundle being in the Sobolev space $H^{k,r}(E)$., it seems that Jost doesn't use partition of unity to define the Sobolev spaces. He merely requires the existence of an atlas with the property that on compact sets all coordinates changes and all their derivatives are bounded. Then is it possible that we can take two different atlases from the same differential structure such that both atlases satisfy the desired property but define two different Sobolev spaces? That is, the Sobolev spaces depends on the choice of atlases?