I created a staging site using WP-Staging, I do some changes to the Toolset custom post structure, now I need to deploy it but I don't want to deploy all testing data I did from staging site, may I confirm whether I only deploy those tables with _toolset_ are the new update to all Toolset relation setting (please refer to the screenshot) ? Or you suggest me to use Toolset Module migration? Please advise how I can correctly deploy all new development using Toolset.
We are not qualified to answer that question as you need to double-check with the WP Staging plugin support.
Deploying only _toolset ones will not do the trick as Toolset uses native WordPress functions for many parts and normal WordPress database tables are also in play.
"Toolset Module Manager" is a legacy feature for Toolset modules. It will not be suitable for this new version of Toolset context.
I suggest that you have a complete backup of the website and try to move all the data and delete the ones that you do not want.