Hi! I actually called them yesterday and they told me to wait for the email.
She wasn't able to tell me the reason of why there is 2 dates, but she said that if the Oath was actually on August 4 we would receive an email today the latest. If the Oath is for August 10 she said July 31 was the last day to get the email, so she recommended me to call on August 1 if I dont receive the email I imagine that in your case it would be one day earlier so call by July 31 if you haven't received any email
She also said that she thinks that the right date is probably the second (August 10 in my case).