The resultant of two homogeneous polynomials is homogeneous

This theorem is not true. Take $$f=x_0x_1x_2 - x_1x_2x_3 ~text{ and }~ g=x_0x_3x_4-x_0x_4x_5.$$ These polynomials are both of homogeneous degree 3, so according to the theorem the resultant $Res_{x_0}(f,g)$ should be of degree $9$, however $$Res_{x_0}(f,g)=-x_1x_2x_3^2x_4+x_1x_2x_3x_4x_5$$ is of homogeneous degree $5$.

The mistake in the proof in the comments under the question is in determining the homogeneous degree of the $(i,j)$-the entry of the Sylvester matrix, specifically in this part

the $(i,j)$-th entry of the Sylvester matrix is homogeneous of degree $m−j+i$ if $ileq n$, and $−j+i$ otherwise

as it doesn't take into account that certain entries of the matrix are $0$, e.g. if $m=2$ and $n=3$ then at position $(0,3)$ the entry is $0$, but $m+i-j=2+0-3=-1$.

Edit: I believe that the correct homogeneous degree is $$mcdotdeg_{x_0}g+(n-deg_{x_0}g)cdotdeg_{x_0}f$$ because if $f=a_m+a_{m-1}x+cdots+a_{m-k}x^k$ and $g=b_n+b_{n-1}x+cdots+b_{n-l}x^l$ (the coefficients are indexed like so because then the homogeneous degree of $a_i$ and $b_i$ is exactly $i$) then the term $a_m^lb_{n-l}^k$ generically appears in the resultant, up to sign. It may not appear for particular instances of $f$ and $g$, e.g. as in my example above if we switch around $f$ and $g$, but all other terms that do appear have the same degree anyway.