This is really what you need to know.
States of Emergency
The states of emergency are classified as follows:
a) Distress A condition of being threatened by serious and/or imminent danger and
of requiring immediate assistance.
b) Urgency A condition concerning the safety of an aircraft or other vehicle, or of
some person on board or within sight, but does not require immediate assistance.
The pilot should make the appropriate emergency call as follows:
a) Distress ‘MAYDAY, MAYDAY, MAYDAY’
b) Urgency ‘PAN PAN, PAN PAN, PAN PAN.
You can digest Annex 10 volume 2 of the Chicago convention, and all of the ICAO member national variances to your hearts content. A few people get their
knickers in a knot over the subtleties and nuances of each and every aspect. In reality and in practice, it becomes a judgment call for the pilot in command on the day, or it is obvious in accordance with company procedures in the event. One thing that some people need to appreciate is that you are not going to be put on trial, charged, or suffer a fine or penalty because of the honest judgment you make in this respect at the time. You can upgrade or downgrade your initial assessment at any time. The same event may elicit a different response in two different people.
The point of the call or the prefix, is that you are communicating an emergency to somebody else on the radio. Normally this will be to ATC. Sometimes it might be to another aircraft. On occaissions it may be to any unknown station that is monitoring, and has the ability to relay your message and request for assistance.
The initial point is that you are communicating on the radio, so the first thing you do not do is "scream!" That would certainly distort your message, and delay the imperative in what you are seeking to achieve. The second point is that you are the one with the emergency, or the emergency message to convey. It is therefore your assessment that matters, and it is up to you which prefix you feel is appropriate in your given set of circumstances. The person receiving the message will only act on the information you transmit. Whatever protocols they then initiate will be based on your given prefix and assessment.
As an example, imagine an inexperienced private pilot or student pilot becomes uncertain of his position. After trying to orientate himself he realizes he is lost. He knows there is a large international airport in the general vicinity, because his instructor once showed him how to call up for a transit. So he calls up for radar assistance. The controller who is busy with a string of airliners in train, and even more about to be handed over to him, advises that he is too busy to assist at this time. The pilot then prefixes his request with
PAN PAN PAN. Now this gets the controllers attention on an entirely different level. The urgency of the call will result in whatever resources are required to provide the now obvious assistance that is required. The pilot who acted in a timely manner also is able to resolve his difficulty before his problem becomes a potential crisis, such as running low on fuel when inevitably a
mayday call would be required. On the other hand if the weather was worsening, or the pilot felt that other factors were loading his ability, or he was concerned about fuel, then the initial call made might well be a
mayday. There are so many variables that it is impossible to define what is the "proper" or correct prefix in every circumstance.
To take a few of your examples:
Loss of 50% of thrust (read as engine-out)
I think in most cases it would be entirely appropriate to use a mayday prefix. certainly if there was a fire. There are also issues of significant performance loss and workload that warrant as assesment of significant or imminent danger that would be helped by ATC giving you the priority that this situation might warrant. Once the situation had been regularised and the workload reduced some might choose to downgrade the emergency to a PAN. Some might not.
Loss of all but one electric power generator (I believe Batteries are not included as an electric power generator)
This might be time and location dependent. In a situation where a diversion to a nearby airfield was required with one functioning generator and up to 30 minutes of battery backup there is no particular imminent danger. It is possible other factors on a given day, or at a particular location might cause a crew to think otherwise though?
One of the on board people's life is at danger (medical issues)
Again it depends on the circumstances. Certainly if it was necessary to descend and land at a nearby airfield in busy airspace then a Mayday call should normally get the
seas to part for you. On the other hand if the person was receiving treatment or medical attention on board, or a better planned arrival made the need for assistance less time critical in your judgment then a PAN call might be more appropriate. The level of emergency can always be changed as necessary.
Partial or complete loss of any of the primary flight controls
A complete loss of a primary control would almost certainly constitute a situation of serious or imminent danger. A partial loss would depend on the circumstances. Again a judgment call on the day.
Indeed.
Physical damage of the aircraft structure
Well this could be anything from bird strike dent, to a lightning strike, to a broken window, to a panel falling off, to a hydraulic system leak, to an upset, to a collision! It depends on the circumstances on the day. Call it as you see fit, if at all.
Loss of (any?) flight instruments when on IMC
This depends on what, where, who, how. Loss of particular instruments may be of significant importance to a single pilot in a small aircraft. Some instruments of more importance than others. On the other hand in a Boeing 757 with triple backup systems and two pilots, the loss of one or more instruments may warrant little more than a tech log entry, and be nothing to concern anybody else or constitute any sort of emergency.
Shortage of fuel. I don't know a "time figure" that describes "shortage", though. A ball park figure for me would be running on the last 30 mins of fuel, but I'd guess there's a number that states "emergency".
Notwithstanding the initial example I provided, there clearly is a default imperitive for running low on fuel reserves. For airline operations these may well differ, but expecting to reach final reserve fuel (30 mins) before reaching your diversion alternate, would normally warrant ATC's attention with a PAN, and actually reaching the last 30 minutes of fuel, a MAYDAY.
The point I am trying to convey, is that apart from the painfully obvious, the call prefix you use, is based on your judgement at that point in time. It is not based on anybody elses, nor is it open to debate in any way that actually matters. An emergency will be a stressful and busy time for the pilot or crew, and communicating that emergency will help enable other people to properly assist, get other traffic safely out of your way, and hopefully make your resolution of that task easier and more effective. As with anything dynamic, things can change for the better or worse, and time will often allow for more accurate assessments. It is quite acceptable to change, upgrade, downgrade or cancel the emergency status as you as the reporter see fit.