I believe billinaz is referring to local ticketing practices in Arizona. In many states, when people are issued minor tickets, part of the language of the ticket they are given is a court bail document (the part you sign-- the promise to appear). So in essence, you are temporarily in custody or "arrested" on the side of the road and then released on bail your own recognizance. The police officer acts as the court officer. More significantly, it generates an arrest record as if you were taken to jail in certain states (but not in others). This could also be what billinaz is talking about. He should be our resident expert on Arizona LE.
That is false. Being required to appear in court as the result of an traffic infraction is not equivalent to being arrested. Not in any state. It will usually generate a warrant, but it will never generate an arrest record.
A specific set of circumstances result from an arrest -- i.e., the act of law enforcement taking a suspect into custody -- the most visible of which is being read your Miranda rights. This is important in many contexts, for example if you're required to answer "Have you ever been arrested?" after the fact. If you were not read your Miranda rights, and you were not booked into custody, you were not arrested.