I think there might have been an error/typo in these definitions.

1. The Jacobian is the matrix of all first-order partial derivatives of a function (i.e., the generalized partial derivative for any $n \rightarrow m$ function). It's not the partial derivative of a matrix.
2. Therefore, in the definition of the Jacobian matrix $H$, the derivative of function $h$ should be taken, instead of the derivative of $H$ itself ($H$ is the derivative of $h$).