A character with death prophecy won't die unless fulfilling the prophecy.
Mechanically, how do you play it out?
For example, suppose such a character receives a killing wound (via weapon, spell, fall etc.). Does that leave the character incapacitated? Does the unfortunate circumstance never happen (the weapon misses, the stamina roll against the PeCo spell succeeds, the character somehow manages to avoid the fall? Does the troupe (or the player) get to narrate what happens?
there are three factors involved-
the reason for the wound, the nature of the prophecy, and how much the player seems to be abusing the virtue.
You can either contrive to have the prophecy fulfilled in some unexpected way, or you can consider them incapacitated but certain to recover. "You will die at the hands of a prince" may not apply to the guardsman who felled your character, but might well apply to some minor nobleman who fancies himself a doctor in treating your wounds, for example. fallen banners with unexpected symbols are frequently useful, though it is more fair to include such things before the battle if you want this to be a battle where they know they could die (which becomes a question of abusiveness), but it should at minimum protect them from a meaningless death due to a bad roll.
The virtue advises to never contrive to have it fulfilled in some unexpected way after the fact, you have to give fair warning.
"This is a Major Virtue because the character knows he can get away with insane risks; sneaking his prophecy up on him is an unfair way of negating the value of the Virtue."
They are supposed to be taking insane risks knowing that they won't die without a warning, that's the virtue working as intended, not abuse.
You can have the reason be all sorts of things conceptually related to the prophecy, but you always have to warn them before-hand. The nobleman doctor is an excellent example of dying at the hands of a prince, but you should say something along the lines of "hey, this guy counts for your prophecy, are you going to let him treat you and risk him killing you if he messes up or are you going to try and find someone else and stay injured until that happens?" You could mention it before the fight with the guardsman by saying "the nearest doctor is a nobleman and he counts for your prophecy, I would suggest not getting incapacitated if you don't want to risk that guy killing you when he treats your wounds and you are unable to tell him not to" or something like that.
In my troupe I believe it is if you get take an incapacitating wound outside the parameters of your prophecy then you don't die, you're just incapacitated and roll to heal as normal, except you can't die from rolling a 0 or less. We've not had the Death Prophecy guy get hurt that bad so far, but I think that's how we've said it would work.
I certainly wouldn't do it casually, but on the other hand literature is filled with people who built an empire knowing they were effectively immortal until the death prophecy kicked in at the last minute, though generally not without some indication. For example I remember a childhood book about a man who would not die until he saw a mountain upside down and a bird spoke to him. He took over the town, becoming a corrupt sheriff, but had a panic attack when a man moved into town with a talkin bird. At the end of the story he had been flung into the air and was midair upside down when the bird greeted him.
At the same time if they become so averse to their death prophecy that it looks like they will be using it to ward off old age in addition to mundane threats then I will sneak it in on them. I'm also not going to flag every hint.
A Death Prophecy is about storytelling, not mechanics. See ArM5 p.41:
Players may only take this Virtue with the agreement of the storyguide or troupe.
That agreement also includes an agreement about the nuts and bolts mechanics, if necessary.
The most classical death prophecy of them all is told about the Cumaean Sybil.
The lapidary narrative of its consequence also precedes T. S. Eliot's poem The Waste Land:
So many modern people might be able to relate to it.
I may not have explained myself clearly enough. The question that started this thread was not under what circumstances a death prophecy kicks in. It's how it kicks in, concretely, when it does ... in your games.
For example, what happens when a protected character takes a killing blow? Is he still hit but incapacitated? Does the blow instead never connect? Do you subtly steer the character from afar away from death, e.g. he never gets into the duel in the first place? Or do you just allow things to proceed as normal, changing the outcome at the last possible moment, possibly in a spectacular fashion -- headmen's axes that faiil to cut through a neck, funeral pyres that fail to burn a body, characters who survive for centuries entombed, etc.?
We treat it pretty much as Ethan said above: a character having Death Profecy can't die from wounds, so when rolling to see if he heals or gets worse, he can only improve his condition. Also the Dead wound is treated as an extra incapacitating wound, so when it gets improved then it becomes an Incapacitating Wound.
In a visual sense I'd always thought of it as if the character were one of the immortals of Highlander, in terms of recovery.
Ok, so the "last-minute, spectacular" (a.k.a. external-soul-like) approach 
How do you deal with otherwise fatal aging crises? Do they work as if the character made the stamina roll/received magical assistance, recovering by the end of the season?
To be clear I think all the wounds except the Incapacitating one could get worse, there's nothing saying the character can't get messed up really badly by something and spend 3 seasons having to recover like everyone else.
Depending on cosmology (which i change from game to game) I might consider inflicting flaws as a result of "dying" - sure you might not die from that killing blow, but you lost an arm... other possibilities are for them to accumulate decrepitude when they would have died, they may incur warping from whichever realm has given them the virtue, or to simply shrug it off and recover. However whichever mechanism you are planning to use should probably be defined from the beginning.
The one saga I played someone with a Death Prophecy in, I only received a fatal blow once - I tried disturbing a Verditius' wagon, and his watching ward hit me with a Ball of Abysmal Flame. I took what would have been a killing blow, so the SG ruled I was incapacitated and had severe burns. The burns healed without a scar as I received magical healing to boost my recovery.
Ok, how about this?
An elderly Criamon, with a Warping Score of 10, who has reached the last Station on the path of Strife, enters Twilight. He fails his comprehension roll! So, he ought to enter Final Twilight.
But of course, being so far on the Path of Strife, he can't -- Final Twilight is barred to him. In theory, he ought to die on the spot, and dissolve into a mass of adulterations. But there's a Death Prophecy that says he won't die 
Do you send him on a 7+stress die years Twilight, as if he had comprehended Twighlight -- but without him comprehending Twighlight, something not normally possible? Do you assume he comprehended the Twighlight by "adjiusting" that roll? Do you simply assume fate protects him from entering Twilight, and retroactively adjust that roll? Something else?
Personally I would just keep adding stress dice to the number of years for each zero above the "warping score" of 9.
Same, they're gone for years at a time.
If it's an aging thing I would do it as if they had been healed with magic. It means at a certain point they lose every Spring season to illness recovery and one of their stats is absolutely tanked, but at that point the character has had plenty of use out of Death Prophecy.
I don't think Final Twilight is barred to people on the Path of Strife, just that if they do they will spawn horrible adulterations. Hmmmm......sending them off into very long Twilight is a compromise that's an annoyance to the player without screwing over the covenant. Perhaps have them lie there, in agonising pain, as huge numbers of adulterations spawn out of them. When they recover, they can apologise and help their covenant hunt down all the nasty adulterations.
If you really want to make a huge deal of it, give them the Suppressed Gift flaw from Apprentices and remove all their Personality Flaws ( and maybe a Hermetic Flaw or two?) - they are a lovely and wonderful version of themselves with all the adulterations set free, but it has damaged their Gift, and they will need to hunt down all the adulterations before they get their Gift back. If they succeed, they have earned their improved self.
It is quite explicit that they can't enter Final Twilight
"Followers of the Path of Strife can-
not ascend into Final Twilight. They are
so impure that if they attempt to do so,
they die and create terrible adulterations."
HoH:MC p70
HoH:MC p70:
Followers of the Path of Strife cannot ascend into Final Twilight. They are so impure that if they attempt to do so, they die and create terrible adulterations.
Like it or hate it, "they cannot Final Twilight, they die instead" is RAW.
But in theory could someone on the path of strife initiate into the hall of heroes and dwell in the twilight void as a daimon?
In principle, sure. It's not that they have to die. It's just that they can't enter Final Twilight, replacing it with death if they ever get to it.
However! The whole thing about Criamon paths is escaping the prison of time. A reincarnating human has a chance to do it (a Criamon on the path of Strife not in this life, but possibly in a future one). But a daimon is just stuck in the prison of time, with no way out. So, from the Criamon's point of view, that's clearly a losing proposition.
Generally yes, on the other hand someone who was claimed by a criamon parend who decided they did not want to escape the wheel of time could well join the path of strife to avoid final twilight then chose to pursue a daimonic transformation in order to try and stick around.