What's the best video recorder for convertibles? I'm looking for one with as much coverage as possible, reliable, and works with ignition on/off (battery). I love people, but too many lie.

 

I have a VIOFO with GPS. I've not had occasion to access any collisions though, so just video of me driving around the first couple times after I installed it.
Video was great, and there's a "super-capacitor" to keep it alive for G-sensoring for when power is removed so it can record while disconnected should it sense some movement.
I bought a front-and-rear camera system prior, but couldn't find a decent spot for the rear camera for both UP and DOWN roof, so I got this one instead.
https://smile.amazon.com/VIOFO-A119-...s%2A=0&ie=UTF8

 

I've got (almost) the same type of dashcam as Cee Jay has. The Viofo is a good / excellent dashcam for the money.
I've read some issues with it if you use "Parking mode" (record video when parked - engine off) at Dashcamtalk.com.

Possible candidates:
Viofo A119 V3 (single only)
Viofo A129 (single or duo)
Viofo A129 plus (single or duo)
Viofo A129 pro 4K (single or duo)

The Capacitor CJ is talking about is only able to supply power to the device for a very short time, for finalizing a record when power is disconnected for instance.
Dashcams need 5V power, so a converter is needed. Viofo sells "hardwire kits" to achieve this. Mine is connected to the rear of the glovebox using a stripped down cigar lighter plug.