Scale Factors and the Jacobian

Question

Say we have a curvilinear coordinate system with coordinates \begin{gather} q_{1}(x,y,z)\\ q_{2}(x,y,z)\\ q_{3}(x,y,z) \end{gather} with scale factors $h_{1},h_{2},h_{3}$. If we have a vector-valued function of position, $\vec{u}(\vec{x})$, with components $u_{i}(\vec{x})$, we can say that, with sums over repeated indices, \begin{equation} du_{i}=\frac{\partial u_{i}}{\partial q_{j}}dq_{j} \end{equation} This leads to the definition of the Jacobian of $\vec{u}$ as the matrix whose components are $\frac{\partial u_{i}}{\partial q_{j}}$. If we Taylor expand $\vec{u}$ about a point $\vec{a}$ we get \begin{equation} \vec{u}(\vec{x})=\vec{u}(\vec{a})+A(\vec{a})(\vec{x}-\vec{a})+... \end{equation} for some matrix $A$, which is the source of my confusion. I have seen (for example on the Wikipeda article on the Jacobian matrix) that the matrix $A$ is exactly the Jacobian of $\vec{u}$, but should it not include the scale factors? Afer all, $\vec{x}-\vec{a}$ is a vector pointing between two points, not just a difference in coordinates. In particular for a scalar-valued function $\phi(\vec{x})$ we write \begin{equation} \phi(\vec{x})=\phi(\vec{a})+(\vec{x}-\vec{a})\cdot\nabla\phi(\vec{a})+... \end{equation} which in general involves not just the partial derivatives at $\vec{a}$ but also the scale factors at $\vec{a}$ \begin{equation} \nabla\phi(\vec{a})=\frac{1}{h_{1}(\vec{a})}\frac{\partial}{\partial q_{1}}\phi(\vec{a})\hat{e}_{1}+\frac{1}{h_{2}(\vec{a})}\frac{\partial}{\partial q_{2}}\phi(\vec{a})\hat{e}_{2}+\frac{1}{h_{3}(\vec{a})}\frac{\partial}{\partial q_{3}}\phi(\vec{a})\hat{e}_{3} \end{equation} where $\hat{e}_{i}$ is a unit vector in the direction of increasing $q_{i}$. I think the key lies in the fact that $(\vec{x}-\vec{a})$ is a change in position not just a change in coordinates. The expression above for $du_{i}$ doesn't care about what the $q_{i}$ represent, it just treats them as variables of a function and doesn't use any special features of their coordinate nature. I think the relevant thing to consider for such a Taylor expansion would be something like \begin{equation} du_{i}=\frac{\partial u_{i}}{\partial q_{j}}\frac{1}{h_{j}}d\vec{x}_{j}=(\nabla u_{i})_{j}d\vec{x}_{j} \end{equation} where \begin{equation} d\vec{x}=h_{1}dq_{1}\hat{e}_{1}+h_{2}dq_{2}\hat{e}_{2}+h_{3}dq_{3}\hat{e}_{3} \end{equation} and $d\vec{x}_{j}$ is the $j$ component of that. If we wanted to use the Jacobian we should write the Taylor expansion like \begin{equation} \vec{u}(\vec{x})=\vec{u}(\vec{a})+J(\vec{a})(\vec{\Delta q})+... \end{equation} where $\vec{\Delta q}$ is a column vector of the differences in the values of the coordinates between $\vec{a}$ and $\vec{x}$. I am worried that I have some fundamental misunderstanding with how I am thinking about coordinate systems and in particular how finite changes in position relate to the $dq$ and $d\vec{x}$, and what the difference between the $d\vec{x}$ and $dq$ (a change in position vector of a point vs a change in the coordiantes of the point) really is. Is my line of thinking correct? Is the matrix $A$ not just the Jacobian but instead a matrix whose rows each consist of the components of the gradient of a component of $\vec{u}$, including the scale factors? If so how is this usually written?