Keep in mind that when you charge the battery through the camera's USB port, that current is passing through the main board, so if there was indeed a source power mixup (or failure) as JBrew mentions, there is some some danger to the camera. For an abundance of caution, I always remove the batteries for charging. I'm just a big fan of risk mitigation. The visual of plugging my Fujifilm into a 12v cigarette lighter adapter is, well, just downright something I'd rather avoid.
I have both the original Fujifilm charger and an aftermarket set of charger/batteries. The batteries all perform equally, but I find the Fujifilm charger charges about three times faster. This is about an hour-and-a-half for the Fujifilm to charge one battery, and about five hours for the aftermarket charger, although it charges two batteries at once. These times are all with stone dead batteries. And these batteries hold a full charge for a very long time, so I tend to charge all three batteries at my convenience, so I can leave all the chargers home when I go out to shoot.