Well, I've never worried about charging while it's 'on', but usually do it when it's 'off' after I've finished reading.
A few minutes ago my battery symbol showed about 3/4 green, so I just plugged it into a generic 1 amp usb wall charger I have to see what would happen. When it's charging I usually see the green level in the battery symbol moving back and forth between the current level and full, but this time it wasn't doing anything, so I didn't think this charger was working for me. By the time I got my Kobo charger, the symbol was just about "empty" (white), so I plugged in the Kobo charger instead and it showed a little lightning bolt in the middle of the white battery symbol. It remained the same after turning the device off and on again. When I unplugged the charger I got a warning that my battery level was 10% and I should charge it. What happened? Did I damage the battery by trying to charge it with too little current, or break the battery monitor? I'll leave it on the Kobo charger for a couple of hours to see what happens. If I have to do a factory reset to correct this, will I lose any apps I've added and any sideloaded books?