The problem isn't necessarily the water itself but what happens once the water is in your pool, as every pool is unique. If your pool had a higher PH even for a short time it could result in staining. If your neighbor didn't experience the high PH or their iron count was somehow less, that could explain why they didn't see staining.
Regrouting won't solve the problem... it's just a bandaid and the same result can occur. You have to figure out what caused the staining to begin with to prevent it from happening to the new grout.
You should be able to treat the staining without regrouting, and then prevent the problem from repeating.
If Absorbic Acid worked the last time, it's likely from iron. Repeating the process should remove it, and then using sequesterants and monitoring your PH levels to prevent the iron from redepositing in the grout.
Did their water come from the same water supply as your's?