It is actually true. The battery stores just as much energy when it's cold, and has higher efficiency. The problem is that that efficiency drops dramatically

It is actually true. The battery stores just as much energy when it's cold, and has higher efficiency. The problem is that that efficiency drops dramatically as you try to pull more power. A very simplified explanation would be that the charge carriers have a harder time moving quickly through cold electrolyte. This results in a kind of drag (most of which is actually literal drag) that results in a lower voltage on the output. Your phone measures its charge via the voltage (partly), so when it starts pulling more power it suddenly registers much lower voltage. If you turned it off and let the voltage recover, it would show as charged again. Normal lead acid batteries will fail for the same reason; they can't push amps at low temperatures. If they were heated up they would work fine and they will have lost less charge than if they had been sitting at room temperature. Electric car batteries are the same, but the cold temperature enables to push much more power for longer without getting too hot. Please do not compare phone batteries to large, carefully controlled and well-treated batteries like those in new EVs. They are totally different animals. Phone batteries are probably the most brutal of all common battery applications; they are used as heat sinks, have zero temperature control, and regularly experience forces and temperatures that are way out of spec.