ADVI: Option to save family parameters to JSON

Summary:

Currently the mean is saved in the CSV output above the samples, but there is no way to get the estimated (co-)variance out, short of approximating it through the samples.

We could save it to JSON, similar to issues #3181 #3260

Description:

Option to save the non-mean parameters of the family for ADVI. For meanfield this is
https://github.com/stan-dev/stan/blob/9de06d41d0de9201a78c6c3e1de18fbc98496f1d/src/stan/variational/families/normal_meanfield.hpp#L28-L30

For fullrank this is
https://github.com/stan-dev/stan/blob/9de06d41d0de9201a78c6c3e1de18fbc98496f1d/src/stan/variational/families/normal_fullrank.hpp#L28-L31

These could be used to e.g. draw further samples from the approximation from outside the algorithmic run.

Additional Information:

Originally requested on the forums here: https://discourse.mc-stan.org/t/cmdstanpy-variational-how-to-get-density-parameters/37280/

Current Version:

v2.35.0

Start with src/stan/variational/families/normal_meanfield.hpp and normal_fullrank.hpp at the linked parameter definitions, then trace how ADVI currently writes its CSV output and compare issues #3181 and #3260. Done means providing an option that writes the relevant non-mean family parameters to JSON for both meanfield and fullrank ADVI.