Why does Isildur claim to have defeated Sauron when Gil-galad and Elendil did it?

Question

It is well-known that Isildur cut the Ring from Sauron, as in The Fellowship of the Ring, The Shadow of the Past Elrond says:

It was Gil-galad, Elven-king and Elendil of Westernesse who overthrew Sauron, though they themselves perished in the deed; and Isildur Elendil’s son cut the Ring from Sauron’s hand and took it for his own.

Yet in The Silmarillion, Of The Rings of Power and the Third Age it is said:

This I will have as weregild for my father’s death, and my brother’s. Was it not I that dealt the Enemy his death-blow?

Why does Isildur's account differ from Elrond's? No one disputes he cut the Ring off of Sauron.

This question appears to say Isildur gave the death-blow, but doesn't mention the weregild quote, and mostly discusses whether Sauron is killed or not.

Answer 1

Isildur may not be the most reliable narrator over here.

Gollum acquired the ring by murdering his relative Déagol and taking it from him. He would later claim that it was a birthday present.

‘The murder of Déagol haunted Gollum, and he had made up a defence, repeating it to his “Precious” over and over again, as he gnawed bones in the dark, until he almost believed it. It was his birthday. Déagol ought to have given the ring to him. It had obviously turned up just so as to be a present. It was his birthday-present, and so on, and on.
_{The Lord of the Rings - Book I, Chapter 2 "The Shadow of the Past"}

Bilbo got the ring by pocketing it when Gollum had accidently dropped it, and then later claimed that Gollum himself had willfully offered it to him as a prize.

To them his account was that Gollum had promised to give him a present, if he won the game; but when Gollum went to fetch it from his island he found the treasure was gone: a magic ring, which had been given to him long ago on his birthday. Bilbo guessed that this was the very ring that he had found, and as he had won the game, it was already his by right. But being in a tight place, he said nothing about it, and made Gollum show him the way out, as a reward instead of a present. This account Bilbo set down in his memoirs, and he seems never to have altered it himself, not even after the Council of Elrond.
_{The Lord of the Rings - Prologue 4 "Of the Finding of the Ring"}

We see that the ring causes whoever acquires it to embellish the story of how they got it, to make it sound like it is theirs by right, not just by happenstance or otherwise.

So it would fit very well for Isildur to not have been the one to actually kill Sauron, but to still later try to convince himself and everyone else that he did. Note also that he words this as a question, dancing around it rather then saying it outright.

This I will have as weregild for my father’s death, and my brother’s. Was it not I that dealt the Enemy his death-blow?
_{The Silmarillion - "Of the Rings of Power and the Third Age"}

So I'd say to take Isildur's statement with a grain of salt.

Answer 2

From The Silmarillion, "Of the Rings of Power and the Third Age":

Then Gil-galad and Elendil passed into Mordor and encompassed the stronghold of Sauron; and they laid siege to it for seven years, and suffered grievous loss by fire and by the darts and bolts of the Enemy, and Sauron sent many sorties against them. There in the valley of Gorgoroth Anárion son of Elendil was slain, and many others. But at the last the siege was so strait that Sauron himself came forth; and he wrestled with Gil-galad and Elendil, and they both were slain, and the sword of Elendil broke under him as he fell. But Sauron also was thrown down, and with the hilt-shard of Narsil Isildur cut the Ruling Ring from the hand of Sauron and took it for his own. Then Sauron was for that time vanquished, and he forsook his body, and his spirit fled far away and hid in waste places; and he took no visible shape again for many long years.

Tolkien himself never says Sauron had a death-blow delivered by anyone. Elendil and Gil-galad put him into a physical situation where he was vulnerable, then Isildur cut the ring away from him, whereupon he decided to abandon his physical body.

It appears that the Last Alliance as a whole temporarily inconvenienced him, "vanquished" as you say, but nobody can be said to have delivered the killing blow, because he didn't die, what with being a Maia and all. Isildur might not have known that, or maybe he was caught up in the heat of the moment.

Is that not the behavior we have seen from others who come into possession of the One Ring, framing the situation such that their claim to it seems solid? Gollum did it. Bilbo did it. Frodo didn't have to pretend, but he did have to demand it back from Sam. Then Gollum did it again.

Just as characters repeatedly call it precious, they repeatedly try to set their claim such that it seems unassailable.