To start, even though my basic theory of the law disagrees with Highwayman, I know my state has some exceptions to the basic theory and accept he is correct because of his practical knowledge. The idea of it not being a "crime" can square the circle between theory and reality. There is no way to weave a path down through all the laws and reasons why without getting into minute detail that will cover the many ways things can happen within the limitations of the forum.
But, as to why the question was not a stupid one; absent the theory of a traffic infraction is not a crime, a citation indicates an arrest. The signature on the citation is (usually) a promise to appear in court on that arrest. (Or, whatever procedure you have to follow in the state.) In every state I know of, if you don't sign on the dotted line, you can be taken into custody. As long as the officer had probable cause to believe you committed the crime, you can be arrested. (At least, federally.) The common law is a bit different on what is required. There, the law is closely the same between a citizen and a police officer. Many states still hold true on the common law when they talk of a citizen's arrest.
At common law a citizen can arrest a person when a "public offense" has been committed in his presence, when the person committed a felony outside of the citizen's presence and, a felony has been committed and the citizen has a reasonable belief the person committed it. At the common law, the general addition for the police is the *reasonable belief* a felony has been committed and not the limitation as to if it *has* been committed. Obviously, to enforce such rules might have any of a number of penalties and remedies. That would be yet another huge law journal and not our point, so decline. (Nor will I speak of specific common law exceptions like the "shopkeepers" privilege.)
Many states have added by statute to the right of a person or a police officer to arrest; one would be a misdemeanor (aka Not a felony.) can be arrested for if it causes a public disturbance. States have also limited the right for things like "summary" offenses. Others have, by statute, eliminated all citizen's arrests. (None have had their courts agree as far as I know.)
In my state, a police officer could not issue a summons in the situation the OP describes because of how the arrest statutes are described along with all the myriad other exceptions in the law. The officer would turn in a report to the DA (More likely, the city attorney.) who can issue an indictment. Or, he could cite as a citizen's arrest if the person who hit the door is willing to make such an arrest. CdwJava, an officer in my state, can better tell the process or let me know I am wrong. What he says in this thread (Like Highwayman.), I probably agree.