I am on the premise server for about 2 years now. I wonder if it's save now to add field(s) by going to Tools > Advanced > Add new fields. Will it cause any issue to the system functionality and stability?
Also how can I edit the exiting field name in a table?
Any changes to Schema ideally should be followed by the 'Update database structure' step.
You are right, if the 'Yes' is highlighted under Synchronize (within the list of Schemas), it means indeed there was a change to the Schema and that needs to be updated. Complete the Update database structure Wizard and then log off from the Server, login again to view the added fields.
In case no Schema show up as highlighted in that list, or simply if there is a "No" under Synchronize, Update database structure is not required.