I’m going to modify my chart (originally in
this post)
I don’t know that Gandalf told the Elvenking about Bilbo’s ring, though I strongly suspect he did. After the Dwarves’ escape from his forest fortress, he must have
thoroughly investigated the circumstances: any notion that the chief of the guards would deliberately or accidentally free the Dwarves was ridiculous on its face. The Elvenking was both wise and ancient. He must have seen many strange things in his life. It would not take him long to determine that there must have been an invisible thief lurking about his palace, and that this thief was a companion of Thorin & Co. If Gandalf did not at first offer intelligence about Bilbo and his magic ring, I think the Elvenking would ask him if knew anything about an invisible thief, and whether this thief and Bilbo might be one and the same, just to confirm his guess; and if he asked, I think it likely Gandalf would have told him.