As rbp explained, information provided to the pilots is differentiated based on how serious the problem is, from informational up to emergency. Part of designing an aircraft is determining how serious different failures will be.
As technology has advanced, aircraft have become more and more complex. New devices and systems allow aircraft to become faster, safer, more capable, and more comfortable. However, more complexity means more things that can fail. The airplane's systems are analyzed to understand the ways that a system may fail, and how it will affect the rest of the airplane. The results are used to determine how serious different indications may be, and to provide instructions to the crew about what procedures to follow.
With the complexity of modern aircraft, it is much more likely for at least one system to have an issue. It does not make sense to ground an aircraft just because a coffee maker isn't working. Aircraft will have a minimum equipment list (MEL) which allows the aircraft to depart with certain systems inoperative. Circumstances such as ETOPS will further limit which systems may be inoperative. Once in the air, the crew is informed by their checklists about the correct actions to take. Some indications are not critical, others will be. Modern aircraft go even further in determining the severity of indications. Certain less critical information will be suppressed during high-workload phases like takeoff and landing to avoid distracting the crew. Some aircraft will even make it simple for the crew and give them a message like "LAND ASAP".
The basic FAA regulation for crew alerting systems on transport aircraft such as the 767 in your example is 14 CFR §25.1322 (b):
(1) Warning: For conditions that require immediate flightcrew awareness and immediate flightcrew response.
(2) Caution: For conditions that require immediate flightcrew awareness and subsequent flightcrew response.
(3) Advisory: For conditions that require flightcrew awareness and may require subsequent flightcrew response.
It is up to the group certifying the aircraft to convince the FAA whether something requires immediate awareness and/or immediate response.
Some specific alert systems are covered in different sections, such as §25.1305 (d):
(2) A position indicating means to indicate to the flightcrew when the thrust reversing device—
(i) Is not in the selected position, and
(ii) Is in the reverse thrust position, for each engine using a thrust reversing device.
There isn't much more specific than that, but there are other rules that affect the warning system such as §25.729 (e):
(6) Failures of systems used to inhibit the landing gear aural warning, that would prevent the warning system from operating, must be improbable.
So in order to make a failure of the warning improbable, the system may display a false indication rather than fail to indicate a valid problem. Most indications for things such as fires are false. It is much better for the system to assume the worst if in doubt, and for the crew to assume the indication is valid and act accordingly.
So, to answer your questions:
The systems are analyzed to determine which indications are
important enough to alert the crew, and what the crew's response should be. These decisions will be reviewed
at certification to convince the FAA (or local authority) that these
decisions are sound.
As always, the ultimate responsibility of the aircraft lies with the
captain. If the captain feels that diverting is the safest option,
he or she can make that decision. An operator with a good culture of safety will never cause a captain to
fear reprisal for making such an informed decision in the interest