You have a simplexed system - one Logger database on your Rogger. Hence you don't need any synchronization.
You also have an AW database, which you can drop and recreate at any time when the Distributor is running - just run "Initialize Local Database". The AW copies the config from the logger over the real-time feed to the Router.
The HDS is a historical data store that receives information from the logger through the replication processes. As long as the logger retention time is sufficient you could delete the HDS, reinstall it and it will pull the historical data again from the logger, if the logger has not purged it.
You should study the ICM databases. Read your training material again and look at the "Pre-installation Planning Guide for Cisco ICM Enterprise and Hosted Editions"
You have a problem with the installation of the Distributor AW.
The fact that you cannot run "Initialize Local Database" is symptomatic of a basic misalignment between the AW and the Central Controller. You should also see an error like "Cannot initialize the real-time feed" when you try to start the Config tool or the Script Editor. Is that the case?
I suggest running ICMSetup.exe from the icm\bin and going through the setup again. Something is not quite right. I assume your AW is in the ICM domain, you are logged in as the Administrator on the domain, and you are added a Distributor AW.
When installing a AW/HDS, I always suggest that you do this in stages. Install the Distributor AW first and apply the Service Release. Make sure this is working correctly (updateaw shows "waiting for new work") before running ICMDBA to create the HDS database and then running ICMSetup.exe from the icm\bin to make the AW into a Historical Data Store.
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