I doubt lead-acid batteries contain highly concentrated sulfuric acid anyway. The reason you should always add acid to water is to reinforce a safety mindset so when you are working with 18 M sulfuric acid, you don't do the opposite and risk getting yourself hurt. But if you add water to 0.1 M acetic acid ... yeah, nothing will happen.My experience is with large lead acid storage batteries kept at float voltage, therefore in a fully charged state. These batteries have see through cases allowing for views of any boiling or extreme chemical reactions. In maintaining proper electrolyte level no violent reactions were ever observed. Certainly not a lab setting, this is why your statement baffled me.
First you need to understand why water to acid is bad. Dilution of sulphuric acid is a strongly exothermic reaction. Adding a small amount of water to a large amount of acid would generate heat around the diminishing water droplets and could be enough to boil them. The steam could blow acid around.
I don't *know* why the reverse process is safer. It also generates heat, but the nucleus is a diminishing drop of concentrated acid, which if it boils (BP much higher than water), maybe likely to blow water and diluted acid around. Still, I'd pour cautiously, pausing frequently, looking, listening and feeling (indirectly!) for signs of heating. (Not to mention goggles and other PPE.)
Spills - you want to get rid of the acid, so you can hardly do that by adding more concentrated acid. You have to add water. Undesirable and potentially harmful, but necessary. And you add massive amounts of water very quickly, hopefully overwhelming the heating effect with a small quantity of acid.
If there were a large spill, you would need to be very careful. Perhaps use a large hosepipe from a good distance?? (I'really don't know. Just thinking what I might try if I absolutely had to.)
Batteries - Again, you're stuck with the necessity of adding water, so must just try to do it carefully. I think the sulphuric acid gets up to about 4M (but not sure). I can only assume that this is low enough to allow topping up with water, without boiling. One could make a point of adding the water when the battery is at a low state of charge, when the acid is less concentrated.
Adding water to sulfuric acid and adding sulfuric acid to water are equally exothermic. However, adding acid to water allows the heat to disperse throughout the solution so nothing exciting happens. This won't cause acid to spray around.