The actual question is about how anyone knew to call the ring this.
Was anyone actually in a position to know that he attempted to use it to escape the ambush, and then also in a position to see that it betrayed him and slipped off his finger?
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Who knew what Isildur's Bane was? Not Boromir:
'Isildur's Bane is found, you say,' said Boromir. 'I have seen a bright ring in the Halfling's hand; but Isildur perished ere this age of the world began, they say. How do the Wise know that this ring is his?
Boromir probably knew that Isildur took Sauron's Ring, but almost certainly didn't know it was the cause of his death, and didn't know what Isildur's Bane was:
'Isildur's Bane is found, you say,' said Boromir. 'I have seen a bright ring in the Halfling's hand; but Isildur perished ere this age of the world began, they say. How do the Wise know that this ring is his? And how has it passed down the years, until it is brought hither by so strange a messenger?' ...
...'I [Elrond] beheld the last combat on the slopes of Orodruin, where Gil-galad died, and Elendil fell, and Narsil broke beneath him; but Sauron himself was overthrown, and Isildur cut the Ring from his hand with the hilt-shard of his father's sword, and took it for his own.'
At this the stranger, Boromir, broke in. 'So that is what became of the Ring!' he cried. 'If ever such a tale was told in the South, it has long been forgotten. I have heard of the Great Ring of him that we do not name; but we believed that it perished from the world in the ruin of his first realm. Isildur took it! That is tidings indeed.'
...' "This I will have as weregild for my father, and my brother," he [Isildur] said; and therefore whether we would or no, he took it to treasure it. But soon he was betrayed by it to his death; and so it is named in the North Isildur's Bane. Yet death maybe was better than what else might have befallen him.
...'Only to the North did these tidings come, and only to a few. Small wonder it is that you have not heard them, Boromir. From the ruin of the Gladden Fields, where Isildur perished, three men only came ever back over the mountains after long wandering. One of these was Ohtar, the esquire of Isildur, who bore the shards of the sword of Elendil; and he brought them to Valandil, the heir of Isildur, who being but a child had remained here in Rivendell....
So knowledge that Isildur took the Ring and died at the Gladden Fields was not widely known. Knowledge that he died because of the Ring had to have come from the three men who escaped and returned to the North, probably to Rivendell. It's not terribly obvious to me from what's said in LotR that they were in a position to know the details, though. (They were in a battle for their lives in which nearly everyone dies!)
But possibly Ohtar, Isildur's esquire, was with Isildur and witnessed his disappearance and his later reappearance swimming for his life in the Anduin. Even if he didn't know what this meant -- and chances are that Isildur had played with the Ring in Ohtar's presence -- Elrond certainly would have understood.
Additionally, even if Othar didn't bring a specific report back, Elrond may have guessed what happened, since he doubtless knew of the corrupting power of the Ring and its tendency towards treachery, and almost certainly knew Isildur's character and knew that he'd wear the Ring. It's not hard to get from that to a pretty good guess that the Ring betrayed Isildur in some way. And given any eyewitness account of Isildur's death, even the details would be pretty obvious.
As to the specific question, above, "Was anyone actually in a position to know that he attempted to use it to escape the ambush, and then also in a position to see that it betrayed him and slipped off his finger?" the answer pretty much has to be "Ohtar or no one." But even if it is no one, Elrond probably could have made a reliable guess.
In the Unfinished Tales there is the brief story of The Disaster of the Gladden Fields in which the events are told of how Isildur and his sons together with some of the top soldiers of Arnor were wiped out, as well as how the tale made it to Arnor and Rivendell (but apparently not Gondor, unless much later in shortened or garbled form, maybe)
It's mostly written from the POV of the "omniscient author" as it describes events which nobody survived except
a young esquire stunned and buried under fallen men.
He is elsewhere named as Estelmo and he
heard the words of Isidur and [Isildur's eldest son and heir] Elendur at their parting.
Elendur was in Isildur's full confidence as stated in a footnote. Those words in an earlier paragraph are described as
[...] And we bear burdens of worth beyond all reckoning
and later Elendur asks his father whether he couldn't cosplay as Sauron:
what of the power that would cow these foul creatures and command them to obey you?
To which Isildur replies that he found out the hard way that he can't use the Ring and can barely even bear the pain of putting it on. He doesn't have the power to use it and it therefore should go to the Keepers [of the Three Elven Rings]
Later still, Elendur tells dad
Go! Take your burden, and at all costs bring it to the Keepers: even at the cost of abandoning your men and me!
By then, two others of the three overall survivors had already left, with one being Ohtar. One paragraphs says
Isidur turned to his esquire: "Ohtar" [...]
Esquires, according to Wikipedia, before their definition of being more like "gentlemen" (e.g. of the landed gentry) were
personal attendants of Knights of Orders of Knighthood, this title is held by all attendants on the person of the Sovereign,
A young nobleman who, in training for knighthood, acted as an attendant to a knight.
Ohtar would therefore have been quite close to Isildur in daily interaction. Helping to get dressed, shining boots and armour, that sort of thing. A *gentleman's gentleman" as Jeeves would put it. It's hardly beyond doubt that he therefore would have seen the
small case of gold, attached to a fine chain
beyond all doubt [...] had once borne the Ring around Isildur's neck.
Whether Ohtar saw the Ring itself (and somehow resisted the lure due to willpower and out of sense of duty) is possible but speculative and not mentioned in the brief text.
But between Estelmo's and Ohtar's tale (and of course the Wise knowing full well that Isildur has held on to the Ring and it was now gone), those in the know would have pieced together that the malice and draw of the Ring would have brought the Orcs and led to Isildur's downfall. It being his Bane and the outcome of his hubris.
To add to the answers that explain why "it [was] named in the North Isildur's Bane", there was one other case where the name was used: the dream that came to Faramir and (once) to Boromir.
In that case, you may as well ask who sent the prophetic dreams (the Valar? Eru himself?) or how they knew what would happen. The recipients didn't know what it meant, and if Denethor had any idea, he kept it to himself. But the sender of the dream clearly knew the full story, and that the meaning of the name would become clear at the Council of Elrond.
Sauron was defeated in the War of the Last Alliance. With nearly all of his power stored in the Ring, Sauron was vanquished when Isildur cut the Ring from his finger. After the battle, Elrond advised Isildur that he should destroy the Ring, but Isildur decided to keep it. After settling the affairs in the South, he took the Ring with him when he went North.
So Isildur traveled with a large treasure and was attacked. You don't have to know the details of the attack to guess that it happened because of the treasure.
At the very least, Saruman was able to research the fact that the ring was lost in the river. Gandalf tells the story about the last meeting of the Council of Wizards at Elrond's gathering:
'Some here will remember that many years ago I [Gandalf] myself dared to pass the doors of the Necromancer in Dol Guldur, ...
'But we were too late,... Then he [Sauron] gave way before us, but only feigned to flee...
Then for the last time the Council [of Wizards] met; for now we learned that he [Sauron] was seeking ever more eagerly for the One. We feared then that he had some news of it that we knew nothing of. But Saruman said nay, and repeated what he had said to us before: that the One would never again be found in Middle-earth.
"At the worst," said he, "our Enemy knows that we have it not and that it still is lost. But what was lost may yet be found, he thinks. Fear not! His hope will cheat him. Have I not earnestly studied this matter? Into Anduin the Great it fell; and long ago, while Sauron slept, it was rolled down the River to the Sea. There let it lie until the End.'"
Since Isildur was the last known Ring Bearer, it seems pretty obvious that Isildur was the unfortunate soul to lose the ring in the Great River. With the ring being evil and Isildur a great warrior, it seems likely that some historian put the blame of his demise on the known bad guy in the story: The One Ring.