Question about the swords the Hobbits got from the Barrow Downs

Ok, so we know Merry got a Sword of Westernesse desinged for their war against the Witch-king of Angmar, and that's essentially why at the Pelennor Fields, he was able to hurt said Witch King Nazgul enough to allow Éowyn to finish him off.

However, rewind many months earlier and shortly after the Hobbits got said swords from the Barrows, to their first encounter with the Witch-king at Weathertop... Specifically the moment the Nazgul corner Frodo and he puts The Ring on.

Unlike in Peter Jackson's movie adaptatioin, where the Witch-king reaches out to take The Ring from Frodo, and then stabs him when Frodo is able to pull his hand away, but that's the most resistance he was able to put up, in the novel, Frodo actually attacks the Nazgul, even invoking the name of Elbereth Gilthoniel while wearing The Ring, and swings at the Nazgul. It's implied this attack is why Frodo only gets stabbed in the shoulder vs the heart as well.

But in the aftermath, Aaragorn finds the Witch-king's Black Rider cloak and says

‘Look!’ he cried; and stooping he lifted from the ground a black cloak that had lain there hidden by the darkness. A foot above the lower hem there was a slash. ‘This was the stroke of Frodo’s sword,’ he said. ‘The only hurt that it did to his enemy, I fear; for it is unharmed, but all blades perish that pierce that dreadful King. More deadly to him was the name of Elbereth.’

Sooo... my question boils down to, did Frodo simply miss the Witch-king himself, only cutting his cloak? Or did Merry manage to get the only Barrow-blade capable of harming him? Or am I missing something else?

(Also taking a moment to appreciate Frodo, who faced with the Nazgul themselves, in a nearly unescable situation, being overcome by their powers, and putting on The One Ring which was clearly wanting to be recovered and taken back to Sauron in that moment, working it's will against Frodos... and his action is not just to resist, to pull back, or even his attack, but doing so with the invocation of the Elbereth? Pretty badass for anyone, much less a Hobbit... though I do like Peter Jackson giving Sam, Merry, and Pippen attempts to defend Frodo, their (if meager) resistance wasn't mentioned at all in the book)