Distilled is the right water for a battery.
“dead” means different things. If a battery is discharged say by leaving the lights on all night, it is called by most folks “dead”, but it will usually take a charge and soon be good again. If the fluid in the battery is low, which happens over time, the battery will not hold a charge as well as it should, nor deliver the current you need. In that case, topping up the fluid with distilled water to the line will be good for your battery, you should check the level every 6 months or so to insure long battery life. If the cells have flake from the plates accumulated in the bottom, they will short out and the battery is effectively at the end of its first life. A battery shop can remove the plates, clean the cells and restore the battery for a few more years of life, this is a sort of battery ressurection. If the specific gravity of the fluid, the acidity is too delute, the fluid can be changed out to bring the battery back as well. so you see the term dead battery covers a lot of ground.