Used before the first of two or more coordinates or clauses linked by or: no paraphrase with any is available.
WHETHER (conj.) 3. Either: He passed the test, whether by skill or luck.
https://www.ahdictionary.com/word/search.html?q=whether
However OR reads
Used to indicate the second of two alternatives, the first being preceded by either or whether.
Aren't the three excerpts contradictory?
Which one is right?
It is hard to answer your question because you are comparing definitions of “whether” and “or” and a usage note on “either.”
The usage note that you are referring to reports a recommendation that, when “either” is used as a pronoun or adjective, it should refer to a dichotomy, just two alternatives. If there are more than two alternatives, then “any” should be used.
The note continues that many “reputable writers” have ignored that recommendation and that, in any case, it cannot apply to the use of “either” as a conjunction. In short, it suggests that the recommendation has a narrow application and is not massively followed even when applicable.
Yes, there is an apparent inconsistency within the definition of “or.” When not preceded by the conjunctions “either” or “whether,” the definition specifies that “or” is used to link a series of alternatives whereas, if preceded by the conjunctions “either” or “whether, “or” is used to link a pair of alternatives.
Now it is clear from the usage note that this dictionary recognizes that the conjunction “either” can precede a series of more than two. As Jack O’Flahert’s answer says, the definition does not preclude a series of alternatives, and so it is perhaps better to say that the definition is incomplete rather than truly contradictory.
I have a suspicion that what created the apparent inconsistency was a different issue, namely the difference between “inclusive or” and “exclusive or.” If that suspicion is correct, the thought that led to incompleteness was really the distinction between “a” and “not a,” which clearly is a pair of alternatives rather than a series. If that is true, they really needed a usage note to distinguish between inclusive “ or” and exclusive “or” rather than try to convey it in a definition.
I don't think they are contradictory, the last cited definition just doesn't cover all cases. To add to an example from the AHD definition:
I didn't know whether to laugh or cry.
I didn't know whether to laugh, or cry, or punch you in the nose!
Similarly for the example with "whether":
He passed the test, whether by skill[,] [or] luck or cheating, I don't know.
