Problem with battery fires/explosions is that they are NOT fires. They don't go out if you douse them in water, they don't go out even if you starve them of oxygen, and they get hot enough to melt concrete and steel. They move in slow motion (thank God), but they're near unstoppable until burned out.
So if you get a battery fire in your pants, it'll even be slow to heat up, but it'll burn right through your leg if you don't remove it. (or if you can't, because it melted right into your ...)
Also the fumes are toxic and inhaling it will cause "sudden unconsciousness and death".
If you have a chemical fire at home, first GET OUT OF THERE. Don't do anything. Except perhaps rapidly throw off a piece of clothing that got into contact or has melted into the fire.
If you absolutely must (don't), open the windows (ideally from the outside, with a rock), ideally at least 2 of them, and get out of there.
It's thousands of degrees, so water will make it explode, as will any liquid that can rapidly vaporize. Water will also spread the toxic fumes 10.000x faster than they were spreading before you added water.
The water in the air or water that's somehow gotten near the battery might be enough to make it explode if it's badly engineered, so treat it like it might explode or start sputtering 2000 degree droplets at any time without warning (potentially into your eyes).
Don't try to cover it. There's no use. It doesn't need oxygen, so it won't be put out, and just about anything ignites above 400-500 degrees, so you'll just create new sources of toxic fumes.
Frankly in that youtube movie that initial venting from the battery had gone near his body, we'd be talking weeks in the hospital. If it had gone into his face the video would have suddenly gone silent. That was incredibly irresponsible.
Even that's a bit generous. At best I'd describe it as "caught on fire", but not what most people have in mind when you say something "exploded"