What is a crime where prosecution rests with the plaintiff and a crime subject to official prosecution?
Most crimes are so-called indictable offences, i.e. crimes subject to official prosecution, in which case the police may investigate them and the prosecutor may prosecute, even if the plaintiff does not demand punishment for them. Some less serious crimes are offences in which the prosecution rests with the plaintiff, in which case the police may only investigate if the plaintiff has reported it to the police, or the prosecutor, and demands punishment.
If the plaintiff cancels his/her punishment demands during the preliminary investigation, the police will stop the investigations. If it is considered necessary, the prosecutor has the right to prosecute for certain crimes, even if the plaintiff has not demanded punishment.