Gastritis Support Group
Gastritis is a medical term for inflammation of the lining of the stomach. It means that white blood cells move into the wall of the stomach as a response to some type of injury. Gastritis has many underlying causes, from infection with the bacterium H. pylori, bile reflux, or excessive consumption of alcohol or certain foods or drugs like aspirin.

We just bought a filter and put it on our faucet. For me, it sat better with my stomach but I have noticed that since the gastritis has healed, it seems to matter less and less. I would suggest trying to drink whatever seems to make you feel well. (I.E. pick up a bit of filter water or bottled water and see if you notice a difference. If not, just keep going with what you were doing before).
As for Distilled water, again much controversy over whether it's safe or not. That goes for purified water as well. I would say any reliable source of water is better then any carbonated beverage.
I managed carbonated mineral water when I was first sick but have since given my supply to friends and drink my rain water.