Generally Fir is registered only when there is physical hurt to any of the passenger due to one of the drivers negligence. If there is no physical injury then the police can give panchanama report and is issued to the both parties which will be useful for insurance claims .
When a 'major injury' (aacording to the police) happens to person or persons in one of the vehicles only police is blind as to whose fault it was. They will file FIR against the driver of the other vehicle. If the collision was between a car and a two-wheeler they will hold the car driver responsible