harry potter - How do the spheres with recorded prophecies appear in the Department of Mysteries?

Option 2 can be almost certainly ruled out.

Only Dumbledore knew the content of the whole prophecy, and he most likely didn't report it. I say only Dumbledore, not Dumbledore and Trewlaney, because it is known (from PoA) that, after a Seer enters a trance, she can't remember the content of the prophecy:

Harry sat there, staring at her.
"Is there anything wrong, my dear?"

"You -- you just told me that the -- the Dark Lord's going to rise
again... that his servant's going to go back to him.
Professor Trelawney looked thoroughly startled.

"The Dark Lord? He-Who-Must-Not-Be-Named? My dear boy, that's hardly
something to joke about.... Rise again, indeed --"

'But you just said it! You. said the Dark Lord --"

"I think you must have dozed off too, dear!" said Professor Trelawney

Now, Dumbledore could have told the MoM about the prophecy, but seeing that he hired Trewlaney in order to protect her, and given his attitudes towards

1. the Ministry:
   as Professor McGonagall tells Dumbledore in GoF,
   
   

   
   

   "since when did you accommodate the Ministry?"

   
   



2. prophecies
   
   

   
   

   Do you think every prophecy in the Hall of Prophecy has been fulfilled?

   
   

we can be pretty sure that Dumbledore didn't tell the MoM about the prophecy.
Moreover, if someone had actually told the prophecy to an Unspeakable at the DoM , Voldemort could have found him - for instance,through Augustus Rookwood. Note that it is apparently possible to break through Memory Charms via the Cruciatus Curse, so Voldemort could have tortured the Unspeakable in order to retrieve the content of the prophecy.

I think it is safe to assume that nobody told the Department of Mysteries about the whole prophecy.

Option 3 can also be ruled out. As we know, only two people knew the content of the whole profecy, while Snape only overheard the first part. Note that Snape had apparently no reasons to record the prophecy in the Department of Mysteries.
Since Voldemort knew about the first part, but not about the second one, and since he was trying to get the prophecy because he

ha[d] been determined to hear that prophecy in its entirety

the prophecy should have been recorded in its entirety. And since no witness apparently heard the whole prophecy, and nor Dumbledore nor Trelawney told the MoM about it, we can be pretty sure that nobody told the DoM about the prophecy, not even about the first part.

This leaves Option 1: the prophecy is somehow automatically recorded as soon as it is spoken.
If you think about it, it makes sense. We know that a Seer almost immediately forgets the content of the prophecy once he/she gets out of the trance. Would the DoM risk losing a lot of prophecies that a Seer may be obtaining without witnesses (for example during practice at home)? I guess not. A way to avoid this problem would be to automatically detect that a prophecy has been made and record its content.
We don't have any canon information about how this can be done, but I suspect that this may be related to a variant of the Trace. The Trace we know can be applied on the wizard/witch by someone physically close.

Seers are very rare, and they (mostly) belong to well-known families, so they are not hard to find. Seers, or witches/wizards of Seer descent, may be applied a special Trace able to alert the DoM about when a profecy has been made - much like a standard Trace - and also about the exact content.
The standard Trace can't record the exact content of what is being told, but that is not its purpose. Therefore, it may exist a special form of Trace which is specifically made in order to record prophecies, and which may be able (by design) to record what a Seer says after entering a trance.

TL;DR I think it is Option 1 - the prophecy is automatically recorded. How? A special Trace may be applied on Seers in order to record what they say after entering a trance.