Coach and trainer are common, and I know of no restriction concerning which coach. Any team personnel may help an injured player off the field. If necessary, the squad will come out onto the field.Isn't the head coach the one allowed to go attend to the player? Or is it any coach? (At least for most injuries).
The objective is to remove the injured player safely from the field as quickly as possible and resume play. If that takes 9 people, so be it.