Regarding the AA, AAA, C and D batteries. When my grandson was here, we were looking into a drawer with three of the four mentioned, no 9Volt. They were all just tossed into this little drawer. He told me then about these three (AA, AAA, D) igniting, so yes, I would say keep them in their original packaging even if it is open. Take out only what you need and then perhaps reseal the packaging w/tape or rubberbands.
Regarding the 9Volts that ignited in the video: Recall he said that he'd changed the batteries in his smoke detectors and discarded them. At the time change (Spring and Fall), we are to put in fresh batteries. I don't think these batteries are dead or the detector would have been beeping. We are just reminded to change at these two time changes, so that we have "alive" or "fresh" batteries in our detectors in case of an emergency.