It's getting a bit confusing, and I may be repeating what's above, but:
Quote:
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But I do not choose now to do what I came to do
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Means Frodo has chosen NOT to throw the ring into the fire. It does not mean he has chosen to claim the ring.
This may be saying that he has taken the descision himself not the throw the ring in (ie he failed, he couldn't do it as well as possibly everyone else on ME), but he is not saying he has chosen to CLAIM the ring as his own. i.e. Frodo could have sat there and waited for Sauron to reclaim the ring. But what it MAY be saying is that he was forced / co-erced into CLAIMING it as his own.
Therefore: Frodo's has a concious descision not to throw ring into fire, but not a concious descision to claim the ring for himself.