different between MVG and joint MVG?
There's no difference in formula:
Try setting the $C=\text{diag}(\sigma_1^2, \sigma_2^2,\ldots,\sigma_n^2)$ in $(1)$ to see how $(1)$ simplifies to $(2)$. So, your formula $(2)$ is just a special case of $(1)$.
In the diagonal case, the elements of the multivariate distribution are jointly Gaussian, which is much stronger than them all just being Gaussian distributed.
Joint Gaussian distribution has a name of its own (unlike other multivariate distributions with identically distributed elements) because &quot;joint gaussian multivariate random varaible&quot; has the very interesting feature that simple MVG doesn't have: that uncorrelatedness implies independence. This is actually something that makes it useful in communications engineering (because rotation and normalization doesn't change joint gaussianness and doesn't introduce interdependencies between components - so, in an IQ receiver, the noise on I and Q are still independent, even after phase correction, for example).