[Inziladun]

I still favor the Celebrimbor version. However, returning to your question of why Olórin would have brought it to Galadriel, if she already possessed Nenya, I have an idea or two.

In the first place, it's possible Gandalf, when he arrived at the Havens in the Third Age, was coming in blind, maybe knowing little or nothing of the Rings of Power. If he'd been given the Elessar to pass to Galadriel, it might not have occurred to him to wonder why she needed it. Also, he made a point of telling her it was not intended for her sole keeping forever, but that she was to give it to a certain person in the future.
I wonder too, if the powers of the Elessar and the Three, though similar in healing properties, were nonetheless subtly different, owing to the gifts and intentions of their respective makers.