This is a mess because of concessions to dangerous air cargo rules. It is a balancing act between what is safe and what is practical. Different airlines are placing this line is slightly different places. 

Safety authorities (ICAO, IATA etc)would prefer a blanket ban on stored energy devices at all in the cargo hold. (Fires in the cargo hold are slow to be detected and hard to extinguish. If you are over the ocean you may have to fly for up to 3 hours with a burning cargo hold before you can land - not a great situation to be in).  

Lithium batteries have a history of crashing planes worse than say lead acid, alkaline or dry cell batteries. Additionally lithium batteries come on scene relatively late when we were more safety conscious than the old days, so incidents (and rules) attracted more attention.

Obviously passengers want to take their devices with them so the compromise has been to take your power-bank/laptop/phone/drill battery on as carry-on - this way if it catches fire (and this has happened) it is quickly noticed and extinguished, unlike fires in the hold.

One of the fundamental problems with lithium batteries are that they are both an ignition source and a fuel source. The electricity is the obvious ignition source and the electrolyte can be highly flammable. All you need is oxygen (found everywhere) and something to damage the case (a careless baggage handler, conveyor belt or fall?). With other battery chemistry the electrolyte is not flammable (sulphuric acid and potassium hydroxide).