The matrix notation for the gradient of a vector field is generally not-that-well-defined, because as you note there are two reasonable matrix encodings (transposes of each other), which are about equally reasonable. Because of that, whenever the notation $\nabla\mathbf A$ as a matrix is used in the literature, the responsible thing to do is to specify explicitly which of the two encodings is meant.
Both of the two representations have features that make them reasonable:
The second one you mention,
$$\nabla A = \begin{bmatrix}\frac{\partial A_1}{\partial X_1} & \frac{\partial A_1}{\partial X_2} & \frac{\partial A_1}{\partial X_3}\\\frac{\partial A_2}{\partial X_1} & \frac{\partial A_2}{\partial X_2} & \frac{\partial A_2}{\partial X_3}\\\frac{\partial A_3}{\partial X_1}& \frac{\partial A_3}{\partial X_2}& \frac{\partial A_3}{\partial X_3}\end{bmatrix},$$
has the contravariant index as a row index and the covariant index of the gradient as a column index, which is reflective of both of those natures.
On the other hand, the first representation in your post,
$$\nabla A =\begin{bmatrix}\frac{\partial A_1}{\partial X_1} &\frac{\partial A_2}{\partial X_1} & \frac{\partial A_3}{\partial X_1}\\\frac{\partial A_1}{\partial X_2} & \frac{\partial A_2}{\partial X_2} & \frac{\partial A_3}{\partial X_2}\\\frac{\partial A_1}{\partial X_3} & \frac{\partial A_2}{\partial X_3} & \frac{\partial A_3}{\partial X_3}\end{bmatrix},$$
has the nice property that if $\mathbf r$ and $\mathbf v$ are column vectors, then
$$\mathbf r^T \cdot \nabla \mathbf A\cdot \mathbf v = x_i \frac{\partial A_j}{\partial x_i} v_j$$
matches what you would expect as a matrix product, i.e. matching the action of the operator $\mathbf r \cdot \nabla = x_i \frac{\partial}{\partial x_i}$.
For an example of this in action in the literature, see this paper of mine: to avoid confusion, it is important to specify both how the matrix representation is chosen, and how it acts via components. It takes an extra line and it adds a whole lot of clarity to the text.