I'm going to take a markedly-different approach from many of the others; hopefully this perspective is useful, too.
Start by looking at the nearest-neighbor spell to augury: aura of life. 30' radius, resistance to necrotic damage, regains 1hp at start of 0hp turn. Very easy, and the only thing really left up to interpretation is "nonhostile."
Most spells are like this: they're very clearly... spelled out, using defined terms. [/rimshot]
Augury is a different type of spell.
Augury is a super-vague spell, with lots of room for GM interpretation, preference, style, &c. And you're a first-time GM, so you feel like you've got little guidance from the spell. You're right, by the way.
There's a reason for that: while most spells are a manifestation of the player character's will, this spell is designed to do something fundamentally different. Augury allows the player character to engage in conversation with the GM. Not the player; the player character.
I want to be clear on this: the purpose of augury (and some other divinations) is markedly different than most other spells. Once you realize that, things fall into place.
At some tables it's perfectly appropriate for a GM to say to the players "hey guys, if you take on this dragon you're all probably going to die." If necessary, this is justified in-fiction as "your characters are noting the number of dried skeletons of previous adventurers, they know stories of Smaug from their childhood, they've trekked through miles of desolate wasteland just to get here." Or it's hand-waved away.
At other tables that would be completely unacceptable. It'd ruin immersion, it'd destroy people's roleplay, it'd cheapen the game. Crossing the boundary between in-world and at-table knowledge/conversation/interaction is forbidden.
Augury explicitly crosses that boundary, in a way supported in the fiction. Augury allows the character to talk (specify course of action taken very soon) and the GM to respond: "that should go well," "uh, I wouldn't do that," "kinda mixed bag," or "meh."
But what about timing, subject, and adjudicator?
My best advice is never to spend more than 30 seconds on an augury. Listen to the question/proposition, lean back, think for five seconds, and give an answer. If your players or PCs want better answers, they can wait (or pay) for divination, commune, or contact other plane.
The DM can be truthful but still evasive. Exact wording is key here.
ANY QUESTION can be handled this way.
So the question as you have it is "who is the leader of the secret cult?"
So as the DM, I have A TON of options BESIDES the dude/lady's name. (Like oh, I don't know, the dark god they are worshipping! definitely the TRUTH!) First, be specific, name your cult specifically. Because this is vague and there's got to be more than one secret cult in the world, or even nearby. I might answer "Samantha" because Samantha is the leader of the nearby nature nudist cult, which has nothing to do with the cult you are actually looking for.
Second, a question such as "Who is...?" can be as much about identity as it is about a name. Identity is a tricky thing. Bruce Wayne's identity is actually Batman. If you asked "Who is Batman?" I might give you "He is vengeance, he is the night, he is the protector of Gotham..." and so on.
If you ask "What is the name of the cult leader?" I might give the name they have been using in the cult, not their birth name. Because this what they consider to actually be their identity.
If your DM is questioning this spell because they think it would be too easy, that's a failure of imagination on their part because, yes, you can do LITERALLY DO ANYTHING WITH IT.
Ask for the birth name of the cult leader, it turns out their mother changed their name because they were on the run from an abusive father, which is part of the reason why they are psychotic and are now running a cult. OR they were a famous murderer/arsonist before, and they changed their name when they came to this town as an adult.
This may lead to some clues as to their identity but it isn't always straightforward.
It can be, but only if the DM wants it that way. I mean you can answer in a cryptic rhyme or omen...that doesn't speak to an easy answer...
I will say, as others have pointed out, the question has to be reworked, because it's supposed to be about a specific goal, and doesn't really work that way here. It's supposed to be for planning ahead for a specific event. You know a battle is going to happen, so you can ask how many spell casters will be on the battle field on the opposite side, or what direction the attack will come from or if the cultists will have traps. The identity of a person doesn't really fit into a planned event (like a wedding, funeral, or battle). It is a goal, but the DM needs to look at it more closely.
Also, there can be some kind of counter to the divination or spells that let you know if someone know if a divination spell is being cast to learn their name or identity. My rule as a DM has always been, if there's a spell out there, there's a counter-spell out there.
Also, you want to look at how close the Cleric is to the god, and also, if divination is in the god's portfolio.
And, you definitely want to look at the personality of the god you are asking. How they view the world will definitely color the answer (other posters have done better with this point than I have!)
Best Answer
These Prediction spells cannot take themselves into account
These spells tell you information about what would have happened if you didn't cast the spell.
Say you plan to go down a road, and ask Augury about it. It tells you Woe. You decide to go down it anyway, but really stealthily. This time the assassins waiting for you don't get the drop on you, you kill them, and take their fancy stuff. You have changed the outcome to Weal. Augury couldn't take your actions resulting from its own casting (like casting Pass without Trace, or grabbing a couple of guards to come with you) into account.
It's obvious why it can't take itself into account - if it had the answer would have been Weal, but because it told you that the answer would have been Woe, etc.
So that's what the caveat means from a narrative standpoint - that the very casting of the spell may change the predicted outcome.
The GM cannot take the players, or the dice (completely) into account
Say the players ask the GM via Augury about a plan to attack a powerful monster. The GM thinks "Hmm, based on how they've described this plan, their general combat tactics, and their current strength, they should be able to take it down". They answer "Weal".
Then the players deviate from their plan massively, and one ends up dying. Or the dice just really don't go their way, and one ends up dying. Oops, I guess the answer should have been "Woe"/"Weal-and-Woe", instead. But since the GM is not omniscient, it wasn't.
In this case, the caveat is a get-out-of-jail-free card for the GM - they answered the prediction spell in good faith and to the best of their ability, but they still won't always be right.