Don't worry about it, many people have years of experience with the DJI products and how they work, plus I will say the manual still appears to be a work in process and certainly does not cover the update process in extreme detail. Remember to read all of the other documents though including the flight battery guidelines.
There is probably about 15 components that might be updated in a firmware update, the battery is one of them. When you install an update you obviously have one battery in as in the one powering the Mavic at the time, that will be updated obviously. When you go out and fly, if you use another battery it may need to be updated prior to takeoff. Its no big deal, they're usually really quick to update. If the release notes mentioned a battery update then you might want to power it on at home with each battery to save the hassle in the field.
I've not looked with the Mavic but it might be writing log files to a hidden folder on the SD Card that will show all this happening. If you are familiar with computers have a look for the hidden folder and read through the logs. They'll give you a bit more of an insight if they're there.