The update rate is the rate at which the group runs and evaluates its state. That means if you have a group set to run at 30 seconds with a trigger, every 30 seconds the state of the trigger will be evaluated and the group will log if appropriate.
So, with the basic settings, when your trigger goes high it will take sometime between 0 and 30 seconds for the first execution, and then will log every 30 seconds after that. Are you saying that you need the first log to occur sooner? Usually if you’re logging at a 30 second rate it doesn’t really matter if it takes up to 30 seconds for the first record, but if it does in your case it may be possible to come up with a more advanced trigger scenario using action items.