Dead Battery Voltage and its Charging
Short answer: Think of your dead battery as the same battery but with very very reduced capacity. If you leave it alone, it shows some voltage. But as soon as you plug it into a device to extract current from it, it gets empty within milliseconds. When you try to charge it, it appears to fill up quickly.
Discussion: There are many failure modes for each type of battery, some leave them at 0V, some cause them to have high internal resistance, some leave them with reduced charge capacity, some make them leaky. These failure modes can occur alone or simultaneously. The two indications that you mentioned, both seem to be caused by a battery having too little capacity.
But even for a good lithium ion battery, the amount of charge available in the battery is not proportional to it's voltage. An Li ion battery delivers most of its charge between 3.4 V and 4.2 V. Below this voltage, it does have some charge, and it will definitely supply current, but its voltage will fall towards zero very quickly if you draw current from it.