There are still sugars in the juice even though there was no Sugar added. It was not freshly pressed promegranite and was probably pasturized and it was probably not organic. I would continue with the water fast. In my personal opinion you didn't break the fast because you will burn the Sugar quickly and be back into ketosis within an hour or so. So I would just look at this as a temporary blip, but I wouldn't do it again or it will interfere with your fast.
Others might disagree, but to me you are still on day 5 on your fast. I'm not sure how to alleviate your mom's concerns. Someone else may be able to help you here with that.
Lying to me is always a good idea but most people here on this website have morals.