Is DAX updating the cache if the update of the table happens via external sources like AWS console, data loader scripts etc. ?
I have found out about the write-through policy but seems that it works only when used with the DAX client programmatically.
No, DAX is not updated unless you explicitly write through DAX which is not possible using the DynamoDB Web Console or any other source which is not running in the VPC and talking directly to DAX.
Is it the same with ElastiCache?
Yes, Elasticache doesn't act as a write through for DynamoDB so you are tasked with both writing to cache and table.