Omens are a funny thing, when you get down to it. In 'actual life', there is a rich history of divining the risks and rewards in the future by reading the signs in the sky, in the entrails of animals or unusual occurrences.
While the omens themselves are mostly very vague, the process of interpretation by professional diviners makes them more specific to the recipient. It has to, right? If the omen remains vague then why would you let someone rifle through the entrails of a lamb in front of you?
Amateur diviners, on the other hand, are more often interpreting their own omens and are a little more vague. A black cat walking across your path is supposedly an omen of bad luck, but what of? At least with magpies there is a handy verse to do the heavy lifting on the interpretation.
One for sorrow, Two for joy, Three for a girl, Four for a boy, Five for silver, Six for gold, Seven for a secret, Never to be told.
But where does the line lie in the pro-am divining stakes? Is there a licensing process? How many times do you need to interpret an omen (correctly, presumably) before people agree you are a haruspex and not just a chatty butcher?
The difference between life and role-playing games, where magic can be real and fairies exist, is that omens may be real. Now, I'm not saying that magic, fairies and omens aren't real in our daily lives, but the jury is uncertain at best. It's fair to say that most of the examples of omens you can find are either where someone observed the omens and it all worked out well or where someone ignored the omens and disaster befell them. There are few recorded examples where an omen was ignored and everything work out fine anyway. Hang on, perhaps that does mean they do work? (Oh, confirmation bias, you monster).
If I decide that omens work in my game (and I'd suggest don't include them if they don't, otherwise it's the equivalent of all the treasure chests being empty), then there are a few things I want to think about. Firstly, are omens vague or specific, like we touched on above? Having been uncertain when I wrote those paragraphs, I now find myself thinking that the pro-am split of omens is quite compelling. You see a cat riding a dog, chasing a mouse and you think "well, that's a good omen", but you have no idea what of. If you know a friendly neighbourhood diviner, they may be able to interpret this into something more specific involving your ongoing personal quests.
The other choice I think I need to make is whether the in-game effects of omens are narrative or mechanical in nature. If it's a narrative effect, then the flow of events will unfold in a way that more closely supports the omen than the 'base case', if I'm the referee then I can do that. With a mechanical effect, we might say that you have advantage on some or all rolls relating to whatever the omen was concerned with, for example.
So now we have four styles of omen:
Vague and narrative effect
Vague and mechanical effect
Specific and narrative effect
Specific and mechanical effect
If I had the time, I'd make a nifty two-axis graph to show there are four quadrants.
I see no reason why you can't mix and match vague and specific in the same setting, on the basis of who is interpreting the omen, at least, but I think that choosing either narrative or mechanical is important. I think it depends on the type of game you like to play, but for me I think I'd go for a narrative effect in almost all circumstances. The gods are nebulous, after all, and giving you "+1 to hit the villain" is too direct for me, it exposes the hand on the control of the universe. The one exception where I would use mechanical effects is in solo games. I could interpret narrative effects, but I think that getting an omen that "your current endeavour will be successful" would work well as an advantage to certain rolls. Solo games tend (in my experience) to be more mechanically driven, even journalling games call for regular rolls to create the world, whereas group games can go for long periods without mechanical resolution, so I think the mechanical advantage for omens feels at home in a solo setting.
By all accounts (well, Wikipedia anyway) in the 7th century BC, King Esarhaddon of the Assyrians, would be informed of some unfolding doom by his diviners. He would install a substitute king to take the impact of the poor fortune and go into hiding. Once it was reported that the danger was passed, he would have the substitute killed and re-assume the throne. If you take your omens seriously, that's exactly the appropriate kingly attitude required in order to prosper.