Around 1:15 in “Newton’s Method for two variables”, it says we can see that Newton’s method, given the formula, can generalise to any n number of variables, with the following dimensions:

to be more graphic:
In a 1D case the gradient tangent line (vector representing just the slope) is used to find the next iteration w/ the (scalar) hessian as second derivative.
But in a 2D (or n-Dimensional) case the gradient tangent plane (or an n-D object/hyperplane) spanned by vectors determines the next iteration w/ the hessian matrix so that you can get to the optimum in an efficient way.

I understand that the dimensions of H^{-1} are named m and n.

you have a 2D optimization case.

My take: we are talking about the subtraction of 2D vectors which match well since the hessian matrix is always a square matrix and so is its inverse (if it exists). So to stay in the convention of the screenshot: n = m . This applies not only in 2D but in general.

In general you can also find a good summary here for both 1D and also higher dimensional cases, @Wenxin_Liu.