"To be elected chairman" vs. "to be elected the chairman"

I have a question regarding the correct use of the definite article "the":

One of my books says:

Definite article the is used before nouns denoting a position that can be held by one person at one time

Example: He has been elected the chairman of the committee.

But another book says,

No article is used before predicative nouns denoting a position that is normally held at one time by one person only.

This means that "He has been elected chairman of the committee" is correct. So my question is this: which one is (more) grammatically correct?

Neither is incorrect, but the usual way to say this is "He has been elected chairman of the committee," as indicated by this Ngram.

I also would have tended to go with that way of saying it, but the Ngram helped me give evidence. I went ahead and checked with "president", "chairperson", "treasurer", etc, and came up with the same results each time. Just in case, I narrowed my search down to comparing "he was elected chairman" with "he was elected the chairman", and got the same result.

So much for popular usage. Let's now take a look at the meaning of "the" in that sentence.

Since "the" normally indicates either that the reader has had previous knowledge of the subject, or that the reader should know that it is the only one of its kind, saying "he was elected the chairman" makes sense. There is, after all, only one chairman, even if you've never heard of him before.

That said, the word "elected" does have its quirks. You might say that "elected chairman" and "elected president", though not at first obviously correct, has come to be a shortcut, since in most committees, clubs, etc, "the" is no longer necessary or helpful to get a meaning across.

As far as the use of definite articles go, both of your examples could be correct, but for the first one you would need to add 'as' as below.

He has been elected as the chairman of the committee.
He has been elected chairman of the committee.

In the later example the words "elected chairman" are operating together.