I have Acquired Nephrogenic Diabetes Insipidus and I can absorb water...from a 'Zazen' water filter. I don't know if it will work with the various other types of DI, but it's worth someone giving it a try. I'm about 95% thirst free when I drink only this water. The filter is expensive, but the replacement bits cost about the same to maintain as a Brita jug. If it works for you, tell your doctors about it, maybe someone will do a study and find some answers.
I'm not sure what it is that does the trick, the filter has various features, any combination of which may be the answer.
And you can carry water from that filter when you travel as checked luggage. Put it in soft drink bottles to withstand plane pressure changes and rough handling. I often travel with 20 litres of water and have never had any trouble.
I have nothing to do with the company at all, it's just something I stumbled across that actually works. I can't describe the relief of not being thirsty, it's just...amazing. http://www.zazenalkalinewater.com.au