Third-person breaks down into two main categories: limited omniscient, where the writer focuses on one character but tells the story from outside their head, and omniscient, where the writer can flit between characters while telling the story from a supposedly detached perspective.
So, when you’re writing limited omniscient it’s he/she but you’re confined to that character’s perspective and, as with first-person/subjective narration, nothing can be shown on the page that they’re not there to see. However, you could have multiple characters written in this way in separate chapters or sections such as Jonathan Franzen does in The Corrections. So while you limited to that perspective for that segment you can fill out the other characters parts of the story. This is not to say you couldn’t do this in first-person, but when I’ve tried this and tested it on people it seems iffy because of the need for voice differentiation, though I always think Joanne Harris manages to make it work well in her Vianne Rocher books, such as Peaches for Monsieur le Curé. But more commonly a combination of first-person and limited omniscient is used, such as Kelley Armstrong’s book Omens and quite a lot of crime fiction.
Limited omniscient is interesting in its own way because it can either be fixed on a character while being distant or be so close as to be in their head almost as clearly as first-person. Sometimes the prose read with the rhythm of the character’s speech and other times they’re more neutral. Due to being more external that first person it’s generally put under the category of the ‘reliable’ narrator, but I don’t see why this should be. Although the protagonist is not narrating the story it will still skew in their direction and this imbalance surely suggests an unreliable narrator. In instances where the narration is close to the character it might be like telling a story about your best friend versus telling a story about your worst enemy making the protagonist appear more favourable in the story and the antagonist less so, thus creating imbalance, but I don’t make the rules about these things.
So we’ve established that limited omniscient is reliable narration except when it isn’t and that it’s somewhere between first-person and godlike omniscience, so far so vague. Perhaps that’s what makes it so difficult to pin down precisely because it can vary so much from writer to writer depending upon their style. I always thought this was the most interesting thing about limited omniscient narration because of the sheer variety; while first-person has variety it is limited to the character’s voice and usually one character, on the other hand limited omniscient can vary and, despite the name, cover more than one character more easily than first-person.
My suggestion when writing limited omniscient would be to experiment and find where you feel most comfortable writing on that scale because it will not always be the same as where another writer feels comfortable on the scale. Usually when I write in limited omniscient I find myself nearer the first-person end of the scale but that might simply because I generally prefer first-person. Nobody can tell you precisely how to write the voice for your story, only you can find that.