I think the confusion is due to the varied use of the terms attribute (attributive), predicate (predicative), argument (≈complement), and adjunct. Depending on how one wants to use these four terms, one can cut the cake in various ways. One can, for instance, emphasize the distinction between attributive and predicative adjectives, e.g.
Attributive vs. predicative
(1) The fat man is hungry. vs. The man is fat.
In the first sentence, fat is an attributive adjective, and in the second sentence, fat is a predicative adjective. The distinction here concerns whether the adjective at hand appears inside or outside of the noun phrase (NP) to which it assigns its property. An attributive adjective assigns its property to the noun phrase in which it appears (the fat man), whereas a predicative adjective assigns its property to a "subject" expression in which it does not appear (the man).
An orthogonal distinction is that between arguments and adjuncts, e.g.
Argument (≈complement) vs. adjunct
(2) Today was hot. vs. We left today.
In the first sentence, today is an argument (the subject argument), and in the second sentence, today is an adjunct (an adverb). Predicates (was hot and left) demand arguments (today in the first sentence and we in the second sentence), whereas they allow adjuncts to appear (today in the second sentence). The defining trait of an adjunct (as opposed to an argument) is that it is additional information that is not necessary to complete the meaning of the utterance; We left alone is also an acceptable sentence.
The passage in the question is referencing these two distinctions, although the use of terminology is indeed not so clear. The adjective drunk, which the passage states is "not ordinary", might be more accurately designated as "not attributive". In the example sentence, i.e.
(3) He wrote most of his poetry drunk.
the adjective drunk is predicative (i.e. non-attributive) because it appears outside of the nominal expression to which it assigns its property (he), and it is an adjunct insofar as it represents additional information that is not necessary to express a complete thought; the sentence He wrote most of his poetry can appear without drunk (versus, for instance, just **He wrote*).
In the big picture, there are significant differences from one grammarian to the next in the use of these terms. There is therefore some freedom for the student of grammar and linguistics to develop a nomenclature that best fits his or her own understanding of the phenomena at hand.