It all can overlap. Say it is a weekday and the Boro Commander is working and another staff chief has the City Wide. The Boro Commander would respond in his own boro, If the Commander is off that day, than the Citywide would respond. City Wide could also respond, even if Boro is working and responding, his call (doesn't have to). The Boro Commanders are also in the rotation for City Wide. If SI has the duty, they would respond to Brooklyn if Brooklyn Boro was off or unable to respond.
Manhattan has Boro and Deputy Boro, both usually reposnd, especially if high profile, wether they are "on duty or not"
There is also a 4th alarm chief that repsonds on the 4th in addition to the Boro or Tour Commander responding on 3rd. Right now it is the two senior 3 star boro commanders. In that case, they are the 4th Alarm Chief. Same car numbers.
Safety (car 12) and other 12 cars, plus Safety Battalion, are not reponding in a command responsiblity, respond as Safety Chiefs.