Difference between $\mathbb{E}[\mathbf{x} \mathbf{x}^{\rm{H}}]$ and $\mathbb{E}[(\mathbf{x}-\boldsymbol{\mu}) (\mathbf{x}-\boldsymbol{\mu})^{\rm{H}}]$
For the first case, as you wrote, it means the elements are not correlated. Since this is a Gaussian Random Vector it means the elements are independent.
It means that at most only one element of $ \boldsymbol{\mu} $ is not zero.
Since if there were more than 1, the matrix $ \boldsymbol{R} $ wasn't diagonal.


Update

Let's define $ \hat{\boldsymbol{x}} = \boldsymbol{x} - \boldsymbol{\mu} $, namely its centered version.

Looking ta the correlation matrix elements:

$$\begin{align*}
\boldsymbol{R}_{i, j} &amp; = \mathbb{E} \left[ \boldsymbol{x}_{i} \boldsymbol{x}_{j} \right] = \mathbb{E} \left[ \left( \hat{\boldsymbol{x}}_{i} + \boldsymbol{\mu}_{i} \right) \left( \hat{\boldsymbol{x}}_{j} + \boldsymbol{\mu}_{j} \right) \right] \\
&amp; = \mathbb{E} \left[ \hat{\boldsymbol{x}}_{i} \hat{\boldsymbol{x}}_{j} \right] + \boldsymbol{\mu}_{j} \mathbb{E} \left[ \hat{\boldsymbol{x}}_{i} \right] + \boldsymbol{\mu}_{i} \mathbb{E} \left[ \hat{\boldsymbol{x}}_{j} \right] + \boldsymbol{\mu}_{i} \boldsymbol{\mu}_{j} \\
&amp; = \Sigma_{i, j} + \boldsymbol{\mu}_{i} \boldsymbol{\mu}_{j}
\end{align*}$$

Now it is really easy to see what is needed for this element to vanish.