It depends on how the 4-pin connector going into the 7xx is wired. Standard wiring from the factory will charge the battery regardless of if the 7xx is on or off. It can be wired so that the battery is only charged when the unit is on. ( Not sure if this is true for the 788, check the manual ) .
Charging is indicated by the blinking yellow light. It should generally only blink once every few seconds when charging. If you see two, three, four or five blinks separated by an interval it indicates some kind of problem. I believe different hardware versions of the unit might be slightly different in that respect as the meanings are no longer described in the manual. Additionally there is a standard rechargeable AA NiMh battery inside the unit to power the time of day clock. After a few years time this might need to be exchanged, if the unit asks for time of day every time it is started. No hurry though, if it works it works and the unit will keep on recording regardless.
// Gunnar