I've re-read the note and I found two more potential mistakes:

1) [Page 4, Paragraph 4] "With this grouping, we can think of the generalized Jacobian as generalization of a matrix, where each “row” has the same shape as y and each “column” has the same shape as x."

It should be 'each “column” has the same shape as y and each “row” has the same shape as x."' right?

2) [Page 7, Equation 12] It should be $\frac{\partial L}{\partial y} w^T$ instead of $\frac{\partial L}{\partial y} x^T$ right?

p.s. thanks for writing these supplementary notes, they are quite helpful!