Hecuba, I don't know why the OP text uses the word "fable." But it does, and it's clear (to me) that it refers to the "proverb" mentioned in 18:2-3. God directs Ezekiel to stop using that proverb (verse 3).
The effect of stopping the use of that proverb is just as you said in the first paragraph of #2: Ezekiel rejects the idea that the sins of the fathers should be visited upon the sons.
If you'll reread #4, you'll see that that is exactly what I also said (using different words). From God's (and now Ezekiel's) perspective, that proverb is a fable.
Where the KJV uses proverb, Luther uses Sprichwort, which according to Cassell's means "proverb."