Setup:
Let $M^n$ be an oriented manifold and let $$\mathcal{A} := \{(U_i,\varphi_i):i \in I\}$$ be a positively oriented atlas ($\varphi_i:U_i \to \varphi_i(U_i)$ preserves orientation). Furthermore, let $\{\chi_i\}_{i \in I}$ be a partition of unity subordinate to $\mathcal{A}$.
Then for $\omega \in \Omega^n(M)$ with compact support, we define $$\int_{M} \omega = \sum_{i \in I} \int_{\varphi_i(U_i)} (\varphi^{-1})^{*}(\chi_i \omega).$$
Now, my question is this. Exactly what do we mean by $\varphi_i:U_i \to \varphi_i(U_i)$ being "orientation-preserving" here?
My general understanding is that if we have two smooth manifolds $M^m,N^n$ with orientation $\omega_M \in \Omega^n(M): \omega_M(p) \neq 0, \forall p \in M$ and similarly for $\omega_N \in \Omega^n(N)$, we get two smooth oriented manifolds $(M,\omega_M),(N,\omega_N)$. Let´s say we have a diffeomorpism $\varphi \in C^{\infty}(M,N)$. Then we say that $\varphi$ is orientation-preserving if there exists a positive function $f \in C^{\infty}(M)$ such that $(\varphi)^{*} \omega_N = f \cdot \omega_M$
Now, in the setup above, $\varphi_i:U_i \subset M \to \varphi_i(U_i) \subset \mathbb{R}^n$, atleast if we assume $M$ is without boundary. Then, since $\varphi_i$ is a homeomorphism, $\varphi_i(U_i)$ is open in $\mathbb{R}^n$. And we know that for open sets $U \subset \mathbb{R}^n$ we get a canonical orientation (non-zero $n$-form) in coordinates on the form $$dx^1 \wedge \cdots \wedge dx^n.$$
Does this mean that what we are actually saying is that for $\varphi_i:U_i \to \varphi_i(U_i)$ to be orientation preserving, we need that for each $i \in I$, there exists a positive function $f \in C^{\infty}(M)$ such that $$(\varphi_i)^{*} (dx^1 \wedge \cdots \wedge dx^n) = f \cdot \omega_M?$$
Comment: Here, I just view $\mathbb{R}^n$ as a smooth manifold with the smooth identity structure, i.e. $(\mathbb{R}^n,[\operatorname{id}])$.
Also, $(\varphi_i)^{*}$ just denotes the pullback here.
By input from peek-a-boo, the overall idea in my preliminary musings are correct, with some details that need to be corrected. Following peek-a-boo:s suggestion, we should say that what we mean by $\varphi_i:U_i \to \varphi_i(U_i)$ being orientation-preserving in the setup-above, is that there exists a smooth function $f \in C^{\infty}(U_i)$ such that $$(\varphi_i)^{*}(dx^1 \wedge \cdots \wedge dx^n) = f \cdot (\iota_{U_i})^{*} (\omega_M)$$ where $\iota_{U_i}:U_i \hookrightarrow M$ is the inclusion from $U_i$ to $M$.
I might edit this answer more later.