Prologue
"The one with the power to vanquish the Dark Lord approaches... born to those who have thrice defied him, born as the seventh month dies... and the Dark Lord will mark him as his equal, but he will have power the Dark Lord knows not... and either must die at the hand of the other for neither can live while the other survives... the one with the power to vanquish the Dark Lord will be born as the seventh month dies..."
The prophecy that led to the deaths of the Potters was a simple thing, as prophecies go, with distinct references to the who and the how and outright stating of the what. In spite of this simplicity, or perhaps because of it, neither Albus Dumbledore nor Severus Snape outright questioned the completeness of its contents. This was not a failure of caution or intellect on their part per se, but rather a fundamental misunderstanding of how prophecies are made on the part of the wizarding world.
Seers, by and large, tend to live together in small conclaves, supporting one another by dutifully recording each of the others prophecies as they are made. By doing this they help keep all of their family members safe, as there are many situations in which a fake prophecy would benefit an unscrupulous party immensely... and hostages make fine bargaining pieces. It also allows them to gather young Seers to themselves, as 'The Gift' as they refer to it, calls each of them to the others, amplifying when they are together.
They also group together to guard the various studies they made over the years on the nature of prophecies. They do this with extreme levels of proficiency because it is, after all, quite difficult to steal from somebody who not only knows when you are coming, but also who you are, what you will look like at the time, and what path you plan to take. It is also quite difficult to catch and kidnap somebody who knows you will be coming for them before you do.
However, this shroud of secrecy created by the Seers regarding the nature of predicting the future was so long and successful that, at the time of the Potter prophecy, there were perhaps twenty people who even suspected that there was a shroud of secrecy, and there were exactly five non-Seers worldwide who had distinct knowledge of how prophecies work. Four of these people were immortal, whether by accident or design, and were deliberately chosen by the Seers as backup containers of some of the more important and specific prophecies regarding the end of the world and how exactly to avoid it for the perceivable future.
The fifth was a very quiet man who had the peculiar ability of reading the history of an object by touch and discovered the secret quite by accident, when he brushed a Seer while walking down the street. Unwilling to get his memory erased and with a rapidly growing core of paranoia, he resolved to bury the memory once and for all.
Somewhat ironically, this paranoia was the only thing that prevented his memory from being wiped, as the Seers discovered that he would have recovered his memory by reading his own history whenever he touched his own skin, and that he would be hit by a bus later that month anyway while running from people that he would (incorrectly) assume were following him to wipe his mind.
The primary reason for this secrecy surrounding the nature of prophecies was simple- prudence. If it was known in the wizarding world that Seers could produce prophecies on demand, people would get it in their heads that not only did they deserve a prophecy for themselves, but that the prophecies they would get should be good. Nobody wants to think their future is bad, and nobody wants to be the bearer of bad news, so when that news comes in the form of a notarized and accredited prophecy and the future really is bad, the bearers tend to wind up dead.
Another reason the Seers tended to keep their control over the prophecies secret was because of the rather metaphorical nature of most controlled prophecies. It was all well and good to go and get a controlled prophecy from a Seer, but when that prophecy ends up being something like 'If the owl crows like the rooster, the locust will glean the wet ashes' people tend to ask for their money back when the Seer can't answer questions like 'Who is the owl?' or 'What does gleaning the wet ashes mean?'. It also wouldn't help their image if it was discovered that the higher they were on hallucinogens or the drunker they were at the time of the prophecy, the more literal the prophecies were.
The Seers learned to take advantage of this though, by having each of the Seers in their conclave make a prophecy about the same thing at the same time while lightly smashed, so that they could pool the prophecies gathered and parse out what was actually going to happen.
Soon after the policy of secrecy was initiated, uncontrolled prophecies (quite a bit rarer than controlled ones) became the only kind of prophecy heard by your everyday witch or wizard, as true prophecies, whether controlled or not, cannot be Obliviated from the recipients memory.
Uncontrolled prophecies are very different from controlled prophecies.
Controlled prophecies are given very grudgingly by Time to Seers in things unrelated to the Seers immediate wellbeing, and are deliberately metaphorical and speculative due to 'forcing' Time to give them a piece of the future that does not belong to them. Uncontrolled prophecies, on the other hand, are Time screaming 'TELL SOMEONE THIS, THIS IS IMPORTANT' as loudly and literally as possible into the Seers' metaphorical ears. Because Time is screaming so loudly, and the prophecy is coming from the future towards the past, there is an incomplete 'echo' of the prophecy which happens one day before the actual prophecy.
Normally, the other Seers would recognize the uncontrolled prophecy as exactly that due to the incomplete phrases and literal nature of what phrases were complete, and they would wait eagerly for the following day for the complete and whole prophecy. Sybill Trelawney, however, did not have these supporters when she gave the Potter prophecy.
Trelawney, quite taken in her youth by what she had read about her great-great-grandmothers' prophecies, worked extremely hard to 'refine' her inner sight, despite the fact that she had so little of 'The Gift' that she could not actually make a true controlled prophecy. Because of her lack of ability, she was passed over by Seers in the conclaves as a child and told to find occupation elsewhere. Her magic, however, did its best to mold itself to her wishes as she grew, expanding on her tiny shard of 'The Gift' and making her offhand predictions uncannily accurate, although her attempts at true prophecy were still uninspired at best.
Dumbledore and Snape, both totally unaware of the fact that actual prophecies tended to be a bit more complete than what they heard, decided that they had heard the whole prophecy and promptly acted on it. Dumbledore quickly left the interview in order to gather information on whoever had caught the first part of the prophecy, and Snape quickly went to Voldemort and relayed all he had heard. Trelawney , when told by Dumbledore at the abrupt end of her interview that she had made a true prophecy, was so excited that she didn't think to go to the conclaves and report it. She skipped home, happy with newfound confidence in her gift and a new job.
The next day, in the privacy of her own apartment and slightly high from her celebratory bowl of mushrooms, Sybill Trelawney finished her first true prophecy:
"The one with the power to vanquish the Dark Lord approaches. He is born to those who have thrice defied him in battle, born as the seventh month dies under the sign of the lion. As his birth sign the Nemean he will have an impenetrable hide, one part from his mother, one part torn from his enemy, one part forged by his guardian and one part quenched in the flames of the everlasting bird of flame. The Dark Lord will hear this and the Dark Lord will mark him as his equal, but he will have power the Dark Lord knows not, a flame of life and death and magic forged by that flame, ever growing and living in balance. Eventually, the Dark Lord will seek him out and either must die at the hand of the other for neither can live while the other survives. The Dark Lord that dipped himself in the Styx six times to avoid death will nearly lose his grip on shore during the seventh, and the boy will gain all the other had in payment for pulling him to the edge."
So passed the first true prophecy of Sybill Trelawney, unheard and unrecorded.