The Avianca crash comes to mind. The crew said they're low on fuel or something like that, but continued to accept ATC vectors. They were pilloried for not declaring an emergency and out of the accident came the silly "minimum fuel" concept. The crew should have been pilloried for not getting the damn thing on the ground before they ran out of gas. "Declaring" an emergency would only have punted the outcome to ATC. IMO, the emphasis in teaching should be to DO what's required, not SAY what really isn't.
Minimum fuel was not new after Avianca 52. Long before that, Minimum Fuel meant there was not an emergency but undue delay may cause an 'emergency low fuel' situation. After Avianca there were bulletins, education programs, maybe even an Advisory Circular, I'm not sure. Mandatory briefings were done throughout the ATC field reminding controllers that Minimum Fuel was not an emergency. Although it was already a requirement to pass important information from controller to controller concerning a flight, it became 'specifically' required to relay Minimum Fuel reports.
2−1−8. MINIMUM FUEL
If an aircraft declares a state of “minimum fuel,”
inform any facility to whom control jurisdiction is
transferred of the minimum fuel problem and be alert
for any occurrence which might delay the aircraft
en route.
NOTE−
Use of the term “minimum fuel” indicates recognition by
a pilot that his/her fuel supply has reached a state where,
upon reaching destination, he/she cannot accept any undue
delay. This is not an emergency situation but merely an
advisory that indicates an emergency situation is possible
should any undue delay occur. A minimum fuel advisory
does not imply a need for traffic priority. Common sense
and good judgment will determine the extent of assistance
to be given in minimum fuel situations. If, at any time, the
remaining usable fuel supply suggests the need for traffic
priority to ensure a safe landing, the pilot should declare
an emergency and report fuel remaining in minutes.