I haven't found absolute guidance in the official Code. The official Code is concerned with current standards, and not with older systems that were acceptable at one time.
At one time, with fire alarm systems, there were only two signals: alarm and trouble. Now, there are three general types of signals: alarm, supervisory, and trouble.
With monitoring, however, there are two classifications of people that should be concerned with signals coming out of your buildings: Emergency Responders and On-Site Maintenance/On-Site Management Personnel.
The emergency responders, like the fire department, are the ones that will be dispatched when there's an emergency, like a fire. They require a person to call them to report an emergency so they can respond. Usually, they don't want to hear about supervisories, unless they have to respond to the supervisories.
On-Site Maintenance/On-Site Management Personnel
These are the people who are going to respond to supervisories and troubles.
The supervisories they have to respond to are signals from equipment or devices that are not directly part of the fire alarm system. The supervisory signal says that something is not right with what is being supervised.
The troubles they have to respond to are signals from the fire alarm system itself. The trouble signal says that something is not right with the fire alarm system.
Alarm - Supervisory - Trouble Notification
When there's a fire, so they can arrive, the fire department needs to be notified, and the on-site personnel need to be notified so they can take action.
When there's a supervisory, like a duct detector in alarm or a valve in the sprinkler system is turned off, the on-site personnel need to be notified so they can take action to fix the cause of the supervisory.
When there's a trouble, like anything from a bad battery to a complete failure of the entire fire alarm system, the on-site personnel need to be notified so they can take action to fix the cause of the trouble.
Supervisory versus Alarm and Trouble
In one sense, a supervisory is an alarm signal that doesn't evacuate the building or call the fire department. It still requires immediate action by on-site personnel.
In another sense, a supervisory is sometimes (but not always) a more pressing problem than a trouble. A duct detector alarm is a more pressing problem than a bad battery in the fire alarm system.
But then, a trouble may be a more pressing problem than a supervisory. Think of it this way. If the fire alarm panel has a complete failure, that's merely a trouble. The "Merely a Trouble", though, means that a real fire won't be detected, and a supervisory problem won't be detected, either.
The supervisory requires immediate action by on-site personnel, but then, so does a trouble.
The Panel's Display
When responding, the fire department reads the panel to help locate the fire; the on-site personnel should also be trained to read the panel to determine the cause of a supervisory or trouble.
In one case, in an apartment building, I was told the fire department had to be dispatched on alarms, supervisories, and even troubles. In another case, in an elementary school, the fire marshal told me not to dispatch on smoke detector alarms (even though the smoke detectors evacuated the building), but to dispatch only on a waterflow switch or the pull station in the office.
The fire marshal, who has to deal with alarms, decides what will dispatch the fire department. When deciding whether to send the supervisories as an alarm or a trouble, the fire marshal should be consulted.