It sounds as though the shop owner/management decided not to file charges, which is typical in Japan in the case of first offenses. From your description, it sounds as though she was detained by a security guard (not a police officer) at the mall. If her extent of involvement with the actual police began and ended at the local police box level then it is almost certain no official paperwork was done and even more certain that she was never technically under arrest. (Being escorted to a police facility can happen in Japan without there actually being an arrest. Many foreigners who have been taken to a police box or police station think that means they have been arrested, but this is not the case. The Japanese term is 任意同行 (nin'i doukou) which means "voluntary accompaniment")
Since there was most likely no actual arrest there is no need or obligation for the Japanese police to contact base officials or your thieving skunk of a friend's parents. However, there is also nothing to prevent them from doing it anyway. Hence there is no way of knowing about that part until it does or does not happen. As regards the criminal record, she can't have one of those without a conviction. Her name may be in some police database or (more likely) on some piece of paper in a file cabinet in a police box for future reference. After all, the merchants need some way of knowing to go ahead and prosecute her sorry butt the next time she acts a fool and let her learn that Japanese prisons include forced labor, rigid discipline, and extremely limited contact with the outside world.
Please ask your friend to not be such an embarrassment to other law-abiding foreigners in general and Americans in particular. And inform her that in the event she is ever taken into the Japanese criminal justice system by actually being arrested she will learn the following:
1. She can be held for up to 23 days without charges, without contacting anyone, and is not allowed to have a lawyer at any point of the process up until criminal charges are actually filed. Her sole benefit as a foreigner is that the police are obligated to contact someone from the American Embassy so they can come by and check on her well-being, if she so desires.
2. In general, there is no pre-trial release (bail) unless and until the prosecutors have a signed confession in their hot little hands to go to court with.
3. The prosecutors enjoy a conviction rate of close to 99%. (See #2)
4. Japanese prisons suck.