De-ionization process removes all possible salts, ions but not bacteria found in water through a process known as resin. So basically we want to avoid ions and salts in water so why is distillation any different as it does removes the both salt, ions and also bacteria. So what makes you think that de-ionized water is superior to distilled water. basically, once we are talking on context of batteries then bacteria does not says anything to the cells or we can ignore the bacteria factor in lead acid batteries. Only advantage of de-ionized water over distilled water in context to industry is that its cheaper to produce de-ionized water but I don’t know the detailed process of producing de-ioinized water that’s why never proposed anyone to prepare that at home.
Last thing, just wanted to inquire from you. Do you understand what distillation exactly is??? Distillation its self is an expensive and time consuming procedure. Do you think people at factory have funds and time to distilled the water first and then further try de-ionizations once first has already achieved the purpose in an expensive manner but most effective one.
You said, in your university tap water is already distilled. Let me speak and define you what distillation is. Distillation involves, heating the water and it changes its form into gas and then recapturing the gas and condensing it (cooling it) back into liquid. All your minerals will be left in your old container and what you will have a pure water with no minerals and free from ions.
Anyway these are lead acid batteries after all, if you have an access to any of the one and electrolyte is produced with it then you will be far better off than the electrolyte filled by the battery shops.