I've tried to reply to this thread several times. Literary criticism has been a minor part of my education, but it's been a while ago, and the three-point-of-view model has never been my favourite. The following is my attempt to summarise what I remember about how the terminology was used, and in what context. I'm not entirely reliable, selective memory and all, but I'm going to try and give and short summary in as easy terms as I can manage. For what it's worth, I skimmed wikipedia, and it's pretty much in line with what I remember, so that might make my post superfluous, but here I go anyway:
First, imagine a time when fiction was not yet a thing. You have stories, and whenever you hear one, you wonder if it's true or not. If you don't believe it's true, you might wish it was. Sometimes, people will pretend a story is not true, but you suspect it is. It's complex, but stories are fun, and always hover between being true or not.
Then there's fiction. Now we know a story isn't true. Making up stories a respected pastime. It's something writers do. And because they're interesting we're willing to suspend our disbelief. So in what ways do writers contribute to our willing suspension of disbelief?
One method that was popular with early novelists (in English) was the epistolary approach. Nobody is telling us - the readers - a story. We're eavesdropping on private conversations.
But most stories, in those early stages, were told. There's a narrator and s/he's talking to us. And we choose to believe him/her. Why?
An easy answer is: because s/he was there. Basically, we have an eyewitness account. We treat a narrator who is part of the story as we would treat an eye-witness. This was called first-person narration. If the narrator was not there, we have third person narration. Why would we believe such a narrator?
We know the author made up the story, and we confer that authority to the narrator. But "he made it up," is not a valid justification under the willing suspension of disbelief. We must endow the narrator with a special knowledge priviledge. Under the willing suspension of disbelief, thus, "he made it up," turns into "he is omniscient". Omniscience, in this sense, is not a trait of character so much as it is a defense against disbelief.
(Note that another answer to the question isn't that s/he "knows everything", but that he "sees everything". They could have called such a narrator the "omniperceptive narrator", but to my knowledge nobody did. Instead we have a variety of terms: objective narrator, camera-eye view... But those terms all come later.)
The same argument still holds true for third limited. How does the narrator know what goes on in the characters mind? S/he knows, because s/he's omniscient. It's just that his/her omniscience is limited to one character's point of view (at a time). Third limited, in this way, branches off from third omniscient.
But why does it branch off at all? What's point of putting a limit on omniscience?
Well, if the distinction between first and third person originated from the question of "Does it ring true?" the distinction between third omniscient and third limited omniscient originates from the question "Does it feel real?" There is a shift of interest from establishing facts to understanding/living experience. So while, under the old distinction, "third limited" is really a sub-type of "third omniscient", the old distinction is not what governs the new distinction:
The distinction between first and third person was governed by reliability; the distinction between omniscient and limited is governed by narrative distance.
Narrative distince is the perceived distance between a narrator and his/her viewpoint character (if there is one). On the one end of the scale we have the editorialising narrator who has little interest in the inner workings of his/her characters. On the other end of the scale we have a near-invisible narrator who hides behind a lightly editorialised stream of consciousness of his/her characters. How much narrative distance is necessary before we stop talking about third limited? The borders are a matter of controversy, but the concept itself is actually fairly clear.
Note that the concept of narrative distance also applies to first person narration: in first person narration, there can be an "experiencing I", a younger version of the narrator at the time of events presented, whose point of view can take over.
Various levels of narrative distance in first person:
"I should have known better, but I was jealous. I considered him a horrible person." --> "Back then, I thought he was a horrible person." --> "I thought: What a horrible person!" --> "What a horrible person!"
Second person accomplishes pretty much the same thing as third limited, except by inviting the reader to take the role of a character, you make the experience more intense. Generally, I'd say, voyeurs prefer third limited, and role players prefer second person.
Instead of rooting a story in the experience of a person, you can externalise everything, without ever getting any editorialising information or opinions of the narrator. Basically, you leave interpretation to the reader. That's what third objective is: a narrator who, rather than hiding behind a view-point character, hides between observable information.
It's a typology that's not based on logical analysis so much as on the history of ideas as expressed in writing. Third limited is relatively young: it entered the scene in the 19th Century (most people mention Henry James). Basically, people developed the first person vs. third person distinction based on why things ring true - and then, when third limited split off from third omni, people realised that now the focus has shifted to what feels real, and analysis according to what rings true is no longer sufficient.
Third person omniscient and third person limited are the same in how they establish trust in the tale they tell; but they differ in the methods they employ to make things seem real. The key difference lies in how both handle narrative distance between narrator and viewpoint character.
I've long been confused about how BethS uses the terms, but in this thread the penny finally dropped. I hope. She's talking about "third person point of view". That is: if you're talking about your own point of view, it's a first person point of view; if you take someone else's point of view, it's a third person point of view. I may be wrong about this, but viewed like this her posts make sense to me. Under this method of speaking, a third person point of view would fall together with what has been called third limited. The difference to the traditional nomenclature is this:
Traditionally, there's a first person narrator and third person narrator. The terms "third person point of view" are derived from that, and merely mean a point of view that is centred around a third person narrator. It's a way of speaking with a particular history. In BethS' way of thinking, the point of view itself would be a third person point of view, because that's how the narrator would experience it. I, the narrator, am speaking in tongues. I am not relating my own point of view.
Does this make sense?