"All good lithium batteries have a little circuit inside which shuts off the charging current when the battery is fully charged."
Sorry that is not true. It depends on the battery. If it is a single round cylinder cell (like the popular 18650 cells) it probably does not have a built in cut off. Some cell phone batteries have 3 connections instead of the usual 2. The 3rd is a sensor that tells the charge controller in the phone to stop charging. If you are talking about a battery pack with multiple 3 volt battery cells (laptop battery), a charge controller is sometimes built into the battery pack and it would cut off power when the battery is fully charged. In my experience most single cell lithium batteries do not have charge cut off sensors.
"It turns out the cigarette plug USB charger I used and blew up some device, puts out 12 volts, not the 5 volts USB is supposed to have
I don't know if this is common."
It is not common in my experience. Also keep in mind that the device you plugged into it may have overloaded it and cause it to fail and go to 12V. Some devices are designed to work with specific chargers that may look like standard USB plugs but in reality are not. My old blackberry phone would only charge from the blackberry specific chargers. Most other generic USB AC adapters and car adapters that I have would not not charge the phone. Yet the when I used a USB cable to connect the phone to a laptop to download or transfer data it would talk but would not charge, of charge extremely slowly.
The voltage monitor you got will work as long as you watch it during the entire charge cycle. IT would be better if it automatically cut off power to the device. That way if you look away at the wrong time you device would not be destroyed by the failure of cigarette USB adapter.