Enable storing the callback outputs in the persistence storage with the flag

[petar-qb]

This PR is similar to #3144. It solves the same issue but with an additional optional callback flag called enable_persistence.
This PR behaves more like a feature, and the other (original) one (#3144) more like a bugfix.

Description:
Add enable_persistence that when True enables storing the callback outputs in the persistence storage.

[gvwilson]

cc @T4rk1n

[ndrezn]

My understanding is that this PR is parallel with #3144, we either merge this or #3144 (but not both!) depending on whether we want to have this as a flag or a default behaviour.