The short answer is that those 12v Cigarette Lighter plugs are pretty useless...
But this thread has a lot more info:
Also, I know that Marco (Rover67) has a lot of knowledge about this floating around in his brain if you wanted to query him directly.
When I get my fridge, I plan to wire it up with a quick disconnect rather than rely on those cigarette lighter plugs... Which is a bummer because I came across the above thread about a week after I installed a couple of cig plugs specifically for use with my fridge.