In general, if you want to know "exactly how much of X on an average is needed to be consumed to cause death", then the magic search term for Google is "LD50", that is the dose that kills 50% of victims.
I've seen the LD50 for distilled water quoted as 90ml/kg for rats (that is, 90 ml per kg of body mass). I presume that figure comes not from theorising about salt intake, but from giving measured quantities of distilled water to rats and counting how many die.
It's likely of the same order of magnitude for humans. Unlike many "poisons" it's not about subtleties of liver chemistry that will vary by species. What killed the rats was the swelling by osmosis of cells and tissues that they didn't want swelled.
For a largeish human that dose is 7 litres, approx 14 pints, drunk at a sitting. Which is fairly unlikely in a school chemistry lesson. Besides, with effort you could drown in less than that!
The distilled water in your chemistry lab is much more likely to kill you for some reason other than its inherent toxicity. For example if it's stored for a while there could be legionella in the tanks. Probably best not to drink it, just as it's best not to drink any water that may have been sitting around for weeks in an unsealed container.
Definitely don't drink other stuff from the labs that your friend tells you is poisonous, because some of it really is. In particular the one everybody thinks of (because, like the water, it's similar to something they know about): lab ethanol can have methanol as an impurity, and may well have had benzene intentionally added because that's an easy way to get the last 4.5% water out that otherwise cannot be removed by distillation. A mouthful of ethanol isn't normally lethal (although it might be illegal), whereas methanol and benzene are both "properly" toxic in the sense that a few grams will kill you.
By the way, I can confirm that in my school days I obtained the same experimental result you did.