It might be more clear here to start with the intrinsic characterization.
We say the exterior algebra of a manifold $\Omega^*M$ is the algebra generated by wedge products of elements of the cotangent bundle $\Omega^*M=\wedge^*T^*M$, or, equivalently, the set of alternating multilinear forms on $TM$. Differential forms are sections of this bundle.
More verbosely, a $k$-form $\omega$ associates each point $p\in M$ with an alternating map $\omega|_p:(T_pM)^k\to\mathbb{R}$ and this map varies smoothly as we vary $p$.
We can, of course, identify a fiber of $\Omega^*M$ with $\wedge^*\mathbb{R}^n$ (by choosing a basis for $T_p^*M$), but there is no canonical way of doing this even locally.
When choosing local coordinates $x^i$ o a neighborhood $U$, the differentials $dx^i$ induce just such a basis for $T^*M$, allowing us to identify $\Omega^*U$ with $\Omega^*\mathbb{R}^n$ locall, i.e. every $k$-form $\omega$ can be written as
$$
\omega=\sum_{i_1<\dots<i_k}\omega_{i_1\dots i_k}dx^{i_1}\wedge\dots\wedge dx^{i_k}
$$
For $\omega_{i_1\dots i_k}\in C^\infty U$. This frame, however, will depend on the choice of coordinates, subjest to the transformation law $dx^i=\sum_j\frac{\partial x^i}{\partial y^j}dy^j$.
The definition you give is essentially the set of all local coordinate descriptions of the differential form, plus a consistency condition to ensure that they agree.
The way it stands, the placement of Proposition 19.5 is a mistake, because $F^*\omega$ needs to be $C^{\infty}$ before one can take its exterior derivative. To fix this, in Proposition 19.7, replace the justification "(Proposition 19.5)" by "(Proposition 17.10)," and then move Proposition 19.7 to before Proposition 19.5.
I see that Arctic Char has proposed the same solution a while ago. I give it my ringing endorsement.
Best Answer
Properly speaking, Lee is not pulling back $\omega$ but its restriction to $U$. We can define the restriction to $U$ of $\omega$ via $\omega|_U = i^*\omega$ where $i: U \longrightarrow M$ is the inclusion map. So really, this defines $(d\omega)|_U$. You can show that these glue together to properly define $d\omega$.