The dose is what matters. I think you would literally need to be in a water drinking contest to try and get sick....why is that? Well because your body has regulatory mechanisms. ADH (anti-diuretic hormone) is a hormone released to regulate this balance. If for instance you flooded your body with extra-water, ADH would be low (not being released as much). Therefore, water would flow through the collecting ducts of the kidney (only water goes through the channels, not sodium) and you would pee a vey low solute urine (a very clear urine)...this would in most cases help regulate the electrolytes. So no, drinking distilled water would not be much of a problem...unless you were in one of those radio contests where they ask you to drink as much distilled water as you can in 5 minutes or something haha. Without ADH, this would for sure be a factor. Diabetes insidious (not DM sugar diabetes) is when there is not enough ADH being released or not responding in the kidney). This would cause you to pee constantly....SIADH (which can happen for many reasons--typically in older individuals) is when too much ADH is released and you would pee less and more concentrated urine. These two really throw off the electrolyte balance. SIADH victims at baseline have a very low serum sodium level. I would avoid distilled water with this condition because the sodium would drop rapidly!
Good demonstration of osmosis though. I explained this in my Type I diabetes physiology blog. It is a great way to think why diabetics pee so much. Sugar brings that fluid back into the vasculature.
Also, you asked why you would die for the opposite of taking in too much salt. So for distilled water (lets say drinking an insane amount in 5 minutes), the blood cells would swell and burst (like you said) and the surrounding vasculature would be hypotonic. This would lead to vascular collapse....MIs, brain death...and so on due to no effective volume.....almost like if someone is in cardiogenic shock.
I would also add that electrolyte balance in itself could lead to death due to cardiac arrhythmias. Cardiac muscle does not contract properly when potassium, Ca2+ and so on are disrupted. I think the osmosis effect would probably kill you first though. As a medical student, I have seen the effects of potassium imbalance, the most likely to cause V-fib (at least with hyperkalemia).
Thank you for the detailed explanation. Few things I didn't know yet and will look up :) also I will have a look at your blog about the psychology part. Guess what we can learn from this is, not never take part in a radio contest haha. Following you for the very good input