Record all inflated time in the balloon system logbook. The record should show total time the balloon was buoyant, regardless of whether it was held on the ground, tethered, or in free flight.
There is the question of accurate times in the logbook. When I first started logging times I used to log to the minute. So a flight would be 1:02 (1 hour and 2 minutes). If you think about it that is next to impossible to be that accurate. I have found that logging to the tenths of an hour is much easier. Example: I flew for 1 hour and 8 minutes give or take a minute or two. I enter it in the logbook as 1.2 hours.
Let us not forget that you are logging into two different books. One is the pilot logbook and the other is the balloon logbook. I use the same recording scheme for both.
One final point concerning the balloon logbook: it is balloon owner’s responsibility to make sure that all repairs are correctly entered in the logbook.