The Mysterious 10th Bearer.

[Voralphion]

We know flat out 9 bearers of the ring, Sauron, Isildur, Deagol, Smeagol, Bilbo, Frodo, Gandalf, Tom Bombadil, and Sam

When exactly was Gandalf a bearer of the ring.? Unless I'm forgetting something, he made sure he never touched the ring as he was so afraid of it corrupting him. Also, I don't think Tom Bombadil could be considered a ring bearer as he only was in possession of it for such a short time, simply holding the ring does not make you a ring bearer. This would be the same for whoever put the ring on the new chain in Rivendell as they only would have needed to hold the ring momentarily.
As to why the chain was changed, perhaps it was done so it would be more secure and so he would be more mindful of it than when it was on the previous chain in his pocket.