I don't think the WK's declaration of invulnerability had anything to do with Glorfindel's prescience. They didn't exactly move in the same circles, and I think it might be silly to imagine Glorfindel's simple statement would develop the kind of high-profile full-on prophecy status that it had in the movie.
He may have considered himself untouchable out of sheer cockiness, and maybe it was partly due to motivational propaganda from Sauron. Or perhaps it was his knowledge of his own nature, which is more complete than ours, that led him to conclude that he was invulnerable to those in the living realm. I happen to like the feel of this last scenario most. It makes sense that the WK's confidence would come from knowing that there were no more Dunedain--the only line of Men with enough spiritual potency to challenge him. Along with this reasoning, it has been proposed in the past that the Witch-King's words "no living man" might be Tolkien alluding to the dead maker of the dagger of Westernesse as WK's true bane. The dagger, some argue, is what "hindered" him, and allowed for the finishing blow 2 THE FASE.
I apologize if I've duplicated anyone's arguments as I really only skimmed the thread.
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