Whether the injury is minor enough will be determined by the medical examiner. Based on experience, your mother's injury undoubtedly constitutes a minor injury: the other finger joint is fractured, certainly not the injured ring finger, and the other four fingers can all constitute serious injuries.
So, the other party was more seriously injured than your mother.
In terms of sentencing, the sentence for minor injuries is less than 3 years, and for serious injuries is 3-10 years.
If the civil part has been compensated, the criminal penalty can be reduced.
The conflict between your two families is too deep, and there is little hope of reconciliation. Moreover, if the police have intervened, serious injuries cannot be resolved.
Your mother’s behavior was intentional to vent her anger and harm others, and was not considered legitimate defense. During the fight, the other party did not use a knife and did not threaten your mother's life, so it was not self-defense. Your mother committed intentional injury.
Why not give the bailiff a cash gift of 1,000 yuan and assess it as a serious injury?