What safety differences exist between carrying laptops in checked versus carry-on baggage? Recent rumors say that the laptop ban might be extended to flights originating in Europe heading for the USA. This made me think, if laptops are not safe for carriage in the cabin how can they be safe for carriage in the belly of the aircraft? Say that a notebook is short wired such that if it turns on it ignites the battery. Storing the laptop in your carry on luggage would result in a cabin fire which can be put out by the cabin crew. But if this same laptop is in a checked-in bag there is no cabin crew near it. How is this risk mitigated? Or what are the procedures in case of a belly fire?

Furthermore if a laptop is carried inside a carry-on bag it can be switched-on at the security checkpoint to verify that it is a working laptop.
It's not about genuine laptops being safer as checked baggage. They're not, they're safer as carry-on baggage, where eventual fires can be identified, isolated and dealt with. That's why we have rules to make passengers carry-on lithium batteries more instead of checking them in. example

The reasoning explaining the ban is that an attacker might use the shell of a laptop to carry something on board. Turning on the laptop doesn't really prove much about it being genuine. You can take a Raspberry Pi that's size of a matchstick box, pair it with 2 AA batteries to create fake internals and  connect them to the original display. That'll make a gutted-out laptop appear to run for few minutes, enough to fool any quick inspection. Leaving rest of space and mass budget to hide something sizeable, like a bomb or a weapon. 

That's why you can't circumvent the ban by removing the battery and taking a charger instead. It refers to "large electronic devices", not necessary ones with a battery.