Difference between autocorrelations
All these notation should be considered within the context of the development.

So $r_d[k]$ may refer to the auto-correlations sequence of the desired signal whereas the $r_{dx}[k]$ is the cross correlation between the input $x[n]$ and desired signal $d[n]$. So they are two different things.

The latter is used in Wiener optimal filtering to solve the Wiener-Hopf equations (aka The Normal Equations) but I cannot recall where the former is used, can you cite the formula, chapter etc. that makes use of it?