I suppose confusions came from how we define Nabla (or Del or Gradient), \nabla, which is sometimes represented by a row vector, and sometimes by a column vector. Let’s exclude Nabla, at first, and focus on Jacobian.

\textbf{y} =
\begin{bmatrix}
f_1(\textbf{x})\\
f_2(\textbf{x})\\
f_3(\textbf{x})\\
: \\
f_m(\textbf{x})
\end{bmatrix}
=
\begin{bmatrix}
f_1(x_1, x_2, x_3, ...., x_n) \\
f_2(x_1, x_2, x_3, ...., x_n) \\
f_3(x_1, x_2, x_3, ...., x_n) \\
: \\
f_m(x_1,x_2,x_3, ...., x_n)
\end{bmatrix}

Then, Jacobian is as you wrote,

\textbf{J} =
\begin{bmatrix}
\frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \frac{\partial f_1}{\partial x_3}
& ... & \frac{\partial f_1}{\partial x_n} \\
\frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \frac{\partial f_2}{\partial x_3}
& ... & \frac{\partial f_2}{\partial x_n} \\
\frac{\partial f_3}{\partial x_1} & \frac{\partial f_3}{\partial x_2} & \frac{\partial f_3}{\partial x_3}
& ... & \frac{\partial f_3}{\partial x_n} \\
: \\
\frac{\partial f_m}{\partial x_1} & \frac{\partial f_m}{\partial x_2} & \frac{\partial f_m}{\partial x_3}
& ... & \frac{\partial f_m}{\partial x_n}
\end{bmatrix}

To understand what \nabla is, let’s simplify this with m=1. In this case, Jacobian can be written as follows. It is n-dimensional row vector.

f^{'}(\textbf{x}) =
\begin{bmatrix}
\frac{\partial f}{\partial x_1} & \frac{\partial f}{\partial x_2} & \frac{\partial f}{\partial x_3}
& ... & \frac{\partial f}{\partial x_n}
\end{bmatrix}

Then, gradient can be represented as \nabla f(x), which is;

\nabla f(\textbf{x}) = f^{'}(\textbf{x})^{T} =
\begin{bmatrix}
\frac{\partial f}{\partial x_1} \\
\frac{\partial f}{\partial x_2} \\
\frac{\partial f}{\partial x_3} \\
: \\
\frac{\partial f}{\partial x_n} \\
\end{bmatrix}

Yes, this is a column vector, not a row vector. But, many persons write this as a row vector or say does not matter. Actually, that may be right in most of cases. But, again, I suppose confusions come from here.

I believe, from a mathematical view point, if we want to describe Jacobian equation with \nabla, it would be;

\textbf{J} =
\begin{bmatrix}
\nabla f_1(\textbf{x})^{T} \\
\nabla f_2(\textbf{x})^{T} \\
\nabla f_2(\textbf{x})^{T} \\
.... \\
\nabla f_m(\textbf{x})^{T}
\end{bmatrix}
=
\begin{bmatrix}
\frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \frac{\partial f_1}{\partial x_3}
& ... & \frac{\partial f_1}{\partial x_n} \\
\frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \frac{\partial f_2}{\partial x_3}
& ... & \frac{\partial f_2}{\partial x_n} \\
\frac{\partial f_3}{\partial x_1} & \frac{\partial f_3}{\partial x_2} & \frac{\partial f_3}{\partial x_3}
& ... & \frac{\partial f_3}{\partial x_n} \\
: \\
\frac{\partial f_m}{\partial x_1} & \frac{\partial f_m}{\partial x_2} & \frac{\partial f_m}{\partial x_3}
& ... & \frac{\partial f_m}{\partial x_n}
\end{bmatrix}