Suppose we have a differentiable manifold and coordinate chart $M \xrightarrow{\phi=(x, y)} \mathbb R^2$
Let $f: M \rightarrow \mathbb R$ be a $C^\infty M$ function specified "in $(x, y)$ coordinates", meaning we have actually specified $f \circ \phi^{-1}$.
At a point $p \in M$ the tangent vector $\frac{\partial}{\partial x}|_{\phi p}$ properly belongs to $T_{\phi p}\mathbb R^2$. To acquire a corresponding tangent vector in $T_p M$, I suppose it would be necessary to "pullback" this vector through $\phi$ to get the corresponding tangent vector $(\phi^* \frac{\partial}{\partial x})|_p \in T_pM$.
Now if this really is a vector in $T_p M$, then it is a map $C^\infty M \rightarrow \mathbb R$. In this case I should be able to plugin $f$ directly; it should make sense to write $(\phi^* \frac{\partial}{\partial x})|_p (f)$ which reduces to $\frac{\partial f}{\partial x} (\phi p)$. But this does not make sense, because $f$ is not something we can take an $x$ derivative of; $f \circ \phi^{-1]}$ is.
In Wikipedia, the tangent vector $\frac{\partial}{\partial x}|_p$ is defined as an element of $T_p M$ by asserting that its application to $f \in C^\infty M$ is evaluated by replacing $f$ with just the right thing, namely $f \circ \phi^{-1}$. That is, its defined as
$\frac{\partial}{\partial x}|_p (f) = \left ( \frac{\partial }{\partial x}(f \circ \phi^{-1}) \right ) (\phi p)$
How is this definition justified in terms of the pullback (or otherwise)?
The tangent vectors for $T_p M$ are not being pulled-back by $\phi$ but rather pushed-forward by the differential map $d \phi^{-1} : T_{\phi p} \mathbb R^2 \rightarrow T_p M$. The abused notation $\frac{\partial}{\partial x} |_p$ contains a subtle nod to the use of the coordinate chart of which $x$ is a projection, while the push-forward builds in the eventual pullback of $C^\infty M$ functions by $\phi^{-1}$ in the definition of the differential
$d \phi^{-1} := v \mapsto [f \mapsto v(f \circ \phi^{-1})(\phi p)] : T_{\phi p} \mathbb R^2 \rightarrow T_p M$
Which yeilds what Wikipedia offers as a definition of $\frac{\partial}{\partial x} |_p$ by taking it to be $d \phi^{-1} \left (\frac{\partial}{\partial x}|_{\phi p} \right)$