I sometimes check S.I. Hayakawa's book Use the Right Word: A Modern Guide to Synonyms when I want to make sure a word means what I think it means.
I'm not familiar with that work.
I read Hayakawa's book on 'general semantics', lo these many years ago. (I looked him up to make sure it was the same person. I was thinking he represented Hawaii in the Senate, and might have been born there. Actually he represented California, and was born in Canada.)
I wonder if that book was intended to be used in the way you want. Its title doesn't identify it as a dictionary. It seems like more of a style guide than something intended to be an arbiter of 'correct' usage.
In any case, I know of no reason I should regard this book as having equal or superior authority to an unabridged Webster's dictionary.
If I did so regard it, I would still maintain that citing any reputable authority is sufficient defense against an allegation of incorrect usage.
It still applies to the reputation of a doctrine or ideology and is more forceful than retract in suggesting a total disavowing or abject capitulation, including an admission of past guilt and an implied promise to foreswear the error in the future.
I have to wonder if the author meant that use of the word suggests these things in all possible contexts, and that any other usage is incorrect. If he did, that is not consistent with my experience.
In that experience, the word 'recant' is commonly used for a witness retracting a statement. To me it feels like the most apt word, suitable to the gravity of matters that are or may be the subject of sworn testimony.
I think that usage is supported by adequate authority.
I have no quarrel with those who choose to use some other word themselves. I don't agree that 'recant' is technically incorrect, excessive in connotation, or 'loaded'.
If W6 admitted he had lied, and it was actually Zimmerman raining down blows on Martin, recant would apply.
Neither an admission of intentional deception, nor substitution of any particular alternative claim, or any such claim, is required for withdraw/repudiate/retract/recant to apply.