Does a power inverter connected to the 12v battery drain the battery when it's turned OFF?

Depends entirely on the inverter. I once had one that did, which was annoying. I had to add a separate relay in the wiring to the battery to really turn it off. Now I have one that doesn't.

 

A cut off on the line in would provide diagnosis and a fix. Did you fuse it down at the battery?

 

I've been using a Xantrex Prowatt SW since, which I can vouch is genuinely off when it's off. It's a lot more convenient to just have an inverter with a real "off" than to have to fuss with finding and installing a high-current relay or cutoff.

 

It doesn't mean much. Every inverter manufacturer does it their own way. Some have real off switches that completely interrupt all power. Others just put the inverter into a sleep mode which can still draw some power from the battery.

You have to read the manual, contact the manufacturer or break out an ammeter and do a real live test.

Given how small the Prius 12v battery is, it doesn't take much "non-Ready-mode" use to get into trouble.

 

The one I had earlier defined "off" as "please chase the outer segments of your LED display around in an animated circle for entertainment."

 

AND......is nobody going to suggest getting the 12 V battery tested ?
And the charging voltage ?
Except me ??

I think the simple fact that an inverter is in use might mean that the battery has been "abused" and is now worn out.

And I agree that the inverter OFF switch might not turn it completely OFF; a test is necessary.