I was able to resolve my issue. Putting answer here for future reference
I will first list down what I tried:
1. Update Hotfix
2. Add Partial Cache Dependency key
3. Add Cache dependency key for Content caching.
Got an idea by reading answer from this questions:
When I was updating CustomTable A's data, in DB trigger on A would update data in table B which I needed to refresh in Site's cache.
When I tried 'Clear Cache' from Debug application from Admin, it still did not update data in Site.
Also my Custom Table data in Admin was also not getting updated.
So reading one answer from above question, I realized I need to refresh Hashtables for data to be refreshed in admin and subsequently in site.
So I added code to CustomTableForm.aspx.cs in OnAfterSave event handler.
Here I am checking if current CustomTable is my table A, then refresh hashtables of B.