My personal opinion (this is not based on any official guidelines/documents so do not take this as gospel).
Tap water : This is what I use and the only thing I see workshops use to top up. I can only imagine limescale issues if in a very hard water area, and fresh water is being constantly added to the car (to top up leaks or if the cooling system is flushed too often). My logic being that whatever scale is in the water will solidify where it wants to (and might plug tiny gaps), but as long as more water is not being added, more scale is not being introduced to the system so the trace amounts present shouldn't cause any issues.
Distilled water : I wouldn't bother myself. Water is a surprisingly good solvent, but tap water has normally reacted with various trace amounts of various chemicals meaning it is a lot less reactive. I can't say whether distilled water has the potential to cause deterioration of the rubber hoses and seals but I wonder if it is possible. In either event, I can't see the point in spending a lot of money on something that will not give any benefit and could potentially cause major damage.
De-ionised water - I did A level chemistry but didn't do brilliantly at it, and I've currently got brain problems so my memory is a bit sketchy, but I think everything I said about distilled water would also apply to de-ionised water.
The one big suggestion I have for you - your car shouldn't need coolant being topped up, if it needs a top up it must be (slightly) leaking somewhere. Keep a bottle of coolant/water mix in the car all the time in case you need a top up until you are confident the car isn't leaking water, or the leak has been addressed. Just last week I looked at a 116i that needs a new head gasket and probably a new turbo all because of a little dribble from the water cooled turbo.
Now reading through the thread it sounds like you have bought a battery charger, as BMW have advised you stop-start functionality isn't working because your battery is low. The BMW diagnostic computer has a check to bring up frequency of MSA (stop/start) inhibitors and I'm guessing they have done this and found high frequency for low battery.
CTEK is quite a reputable brand and make some good chargers - I believe the one you have purchased has a mode where you can leave it connected indefinitely and it will continually top up your battery if/when required without over charging and damaging the battery. However I don't believe it has any alternator testing functionality. Basic alternator check - check battery voltage with car off, then start car and watch battery voltage. Most modern F series BMW with healthy alternators will be around 14.5v-15.2v when running. If MSA still doesn't work after using the charger, just send the car back to the dealer and let them take care of it under warranty - getting to the bottom of MSA issues can be a piece of cake, but can be a pita as well sometimes.
My personal opinion is that MSA is a daft pointless feature. I remember years ago testing if there was any fuel saving from manually stop/starting a Toyota Carina E 1.6, and I was surprised to find no difference in fuel consumption whatsoever. However stop/start will cause extra wear and tear on the engine, and in particular starter motor. Also if I am driving, I prefer the idea of me being fully in control of a vehicle. Finally, I'm sure most of us at some point have driven a shitty old car that needs a bit of effort to start, but is fine when it's running. Will cause a lot of embarrassment and chaos when a car gets to that stage but still has functioning stop/start feature.
I don't mess around with the MSA system much but one of the most common things I do end up doing with MSA is tweaking coding so MSA is normally off as a lot of people find it annoying.