Every device I've seen - that uses batteries (I have one that plugs into USB and has no battery of its own) - uses lithium Ion Batteries.
Look up these batteries and you'll see that they are better than older batteries - delivering almost as many watts when discharged at high amperage over say 1/2 hour as they deliver with a low amperage over days. The problem is that devices with smaller batteries are asking for 5-10 second bursts of currents that would drain the battery in 1/10 of an hour.
Hrm not as clear as I hoped.
If you have a battery rated for 300mAH - it will happliy deliver 100Ma for 3 hours, It will happily deliver 300Ma for 1 hour, It will happily deliver 600Ma for 1/2 hour - but not all of them are happy to deliver the 2000 - 4000 Ma that a low-resistance coil or a dual-coil setup will require - even if its for just 10 seconds at a time. They may manage it, but die young - perhaps living for 50 recharge cycles instead of the 200 you'd expect under less stressful usage.
The ones made to hold as many Ah as possible - are called high-energy or high-capacity, The ones that hold a bit less Ah, but can deliver higher amperages without suffering - are called high-power or high discharge.
It matters most with the smallest batteries - a 300Mah battery asked to deliver 3000Ma (10X ... 10C) is stressed or needs to use a high-discharge variety. While a 1300Mah battery delivering 3000Ma (2.3C) is not stressing even if its a high energy variety instead of high-discharge.