At the end of this article on tangent vectors, they say each $\frac{\partial}{\partial x_i}$ is a vector field, if we let the point $p$ vary. However, they are only defined in a particular coordinate chart.
I know that $\{ \frac{\partial}{\partial x_i} \mid_p \}$ is a basis for $T_pM.$
However, I also know that vector fields are sections of the bundle $TM$. How can $\frac{\partial}{\partial x_i}$ be a vector field if its domain is not all of $M$?
From the article you linked to:
"Letting the base point vary in the coordinate chart, $\frac{\partial}{\partial x_i}$ are vector fields, but are only defined in this coordinate chart."
The $\frac{\partial}{\partial x_i}$ are vector fields on the domain of the coordinate chart, they are not vector fields on the whole manifold. That is, they are maps $U \to TU \cong \pi^{-1}(U)$ where $\pi : TM \to M$ is the natural projection, and $U$ is the domain of the coordinate chart $(x_1, \dots, x_n)$.
