Ozone is used only as a purifying agent. Ozone decomposes irreversibly in water to form O2 (2 O3 -> 3 O2). That is, when you buy the water, any ozone that was in the water is long gone and therefore you don't need to worry about it.
Distilled and de-ionized water are NOT the same thing. Distilled is boiled and then the steam condensed (as described in the other answer). There is virtually nothing in such water (it will quickly absorb gas from the air and dissolve any residue in the container used to hold it, but in principle it is pure). It contains no ions and nothing else (actually, that isn't even quite true.... it does have H+ and OH- ions in equilibrium, but those aren't contaminants).
De-ionized water, on the other hand, has had all its ions removed (all contaminant ions). Not all water soluble substances ionize in water. So, it is possible for de-ionized water to have non-ionic compounds dissolved in the water (e.g. ethanol). Compare that with distilled which in principle has nothing in it.... They are not the same.