It should be noted, in passing, that use of the word Democracy as meaning merely the popular type of government--that is, featuring genuinely free elections by the people periodically--is not helpful in discussing, as here, the difference between alternative and dissimilar forms of a popular government: a Democracy versus a Republic.
Nonetheless, drawing that distinction is NOT the purpose most people have in using the word, which nine times out of ten is as a less cumbersome way of saying "Democratic form of government, specifically a republic." The writer is correct that the common usage is counterproductive in talking about the narrow topic of the distinction between a popular democracy and a republic, however very few people talk about that topic except to bitch at other people's choice of words.
This is a pet rock of right-wing discussion boards, and about as pointlessly technical as the people who go apeshit on gun boards when some relatively-unsophisticated poster says "Clip" instead of "Magazine."
Since aside from the limited electorate in ancient Athens, there is virtually no historical example of a national popular democracy which even
could have, as Adams put it, 'committed suicide,' it's rather obvious he was using it in the broader sense himself, and not in regard to the very narrow and rare issue of distinguishing a popular democracy from a republic.