What they are saying is that your account is added to the reward table list for that day during the process of logging out, when the system reconciles all of the junk info about your inventory and currencies and all of that.

I think they set it up this way because people will literally rubber band their controller and try to stay online for a week while running in circles. Making a player log out forced a manual event that is hard to fake.

I don't think you need to restart the client. You just need to get into characters on both servers, then let the system log out of that character and to the character menu before flipping servers. This will put both accounts onto the list.