A manifold $M$ of dimension $n$ is certified as such by checking for the existence of a covering $\{U_i\}$ of $M$, each of which is equipped with a homeomorphism $\phi_i : U_i \to \mathbf{R}^n$, called the coordinate maps. If the transition maps between any two charts are smooth, we then $M$ is a smooth manifold of dimension $n$.
Many definitions in manifold theory are local, and involve pulling back by the inverse of a coordinate map; e.g., if $M$ is smooth, then we can define a local derivation on $M$ by $\partial/\partial x^i(-) := \partial/\partial r^i \circ (\phi^{-1})^\ast$(-), where $x^i$ and $r^i$ are local and Euclidean coordinates, respectively.
My question is: if it is so common to pull back along the inverse of a coordinate map, why aren't manifolds defined so that the coordinate maps have image in $M$, rather than in $\mathbf{R}^n$? That is, why don't we instead define a manifold as a topological space equipped with a covering by homeomorphic images, rather than preimages of open sets? Since the coordinate map and their inverse carry the same data, I figure that there is a good reason that I am not thinking of.
I'm happy with a historical answer as well, if there isn't a very compelling mathematical reason for this to be the case.
I do not think that there is a really good reason to define a coordinate map (aka chart) on $M$ as a homeomorphism $\phi : U \to U'$ from an open subset $U \subset M$ to an open subset $U' \subset \mathbb R^n$. [Note that most authors allow arbritary open subsets $U' \subset \mathbb R^n$ as the codamain of a chart, though one can also work with charts having $U' = \mathbb R^n$.]
As you say, $\phi$ and $\phi^{-1}$ carry the same information, thus there is no compelling reason for the above choice of the definition of a chart.
However, writing $$\frac{\partial}{\partial x^i}(-) := \frac{\partial}{\partial r^i} \circ (\phi^{-1})^\ast(-)$$ is a bit sloppy. In fact, for smooth functions $f$ defined on an open neigborhood of $p \in M$ one can define a local derivation at $p \in M$ via $$ \frac{\partial f}{\partial x^i}(p) := \frac{\partial f \circ \phi^{-1}}{\partial r^i} (\phi(p)) = \frac{\partial (\phi^{-1})^*(f)}{\partial r^i} (\phi(p)) . \tag{1}$$ Working with the inverse $\psi = \phi^{-1} : U' \to U$ gives $$ \frac{\partial f}{\partial x^i}(p) := \frac{\partial f \circ \psi}{\partial r^i} (\psi^{-1}(p)) = \frac{\partial \psi^*(f)}{\partial r^i} (\psi^{-1}(p)) .\tag{2} $$ Both in $(1)$ and $(2)$ inverses are involved. It is a matter of taste whether you prefer $(1)$ or $(2)$.
The above definition of charts is certainly used by most authors, but there is also a "minority position" as for example
Especially in the theory of ($1$-dimensional) curves and ($2$-dimensional) surfaces it is quite common to work with local parameterizations which are "inverse charts". Examples:
Do Carmo, Manfredo P. Differential geometry of curves and surfaces: revised and updated second edition. Courier Dover Publications, 2016.
Klingenberg, Wilhelm. A course in differential geometry. Vol. 51. Springer Science & Business Media, 2013.
Shifrin, Theodore. "Differential geometry: a first course in curves and surfaces." University of Georgia (2015): 24.