What exactly does $\frac{\partial(y_1,\dots,y_m)}{\partial(x_1,\dots,x_n)}$ refer to?

I have been asking a rather few questions of this nature lately, maybe I'm starting to realise math notation isn't as uniform as I initially thought it would be...

Question: Does this notation $$\frac{\partial(y_1,\dots,y_m)}{\partial(x_1,\dots,x_n)}$$ refer to the Jacobian matrix $$ J = \begin{bmatrix} \dfrac{\partial y_1}{\partial x_1} & \cdots & \dfrac{\partial y_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial y_m}{\partial x_1} & \cdots & \dfrac{\partial y_m}{\partial x_n} \end{bmatrix},$$ or the Jacobian determinant $\det J$?

This answer seems to support the latter interpretation, while this (and Wikipedia) both support the former.

I am aware of the ambiguity of "Jacobian" being used to refer to either the determinant or the matrix itself, is this a similar case? It's really a bit annoying because when I see things like $$ \left| \frac{\partial(y_1,\dots,y_m)}{\partial(x_1,\dots,x_n)} \right| $$ I don't know if it means the absolute value of the Jacobian determinant, or the determinant of the Jacobian matrix.

Every time that I use it and have seen it, I have used it to refer to the matrix itself. I will typically use det J to refer to the determinant, but I admit that I have used the term Jacobian to refer to both the matrix and the determinant.

This is another case where math is precise, but the language of math is often not. I would recommend that you take care to be specific in your usage - use Jacobian matrix and Jacobian determinant so that there is no confusion.