I get the whole "な-adjectives actually being nouns" deal - but since they modify other nouns attributively they act as adjectives. So it seems natural to me to label the noun (of an な-"adjective") as a noun when it used as a noun and to label the same noun as an "adjective" when it is used as an "adjective". The same also seems natural to me for の-"adjectives"...
Not really. Na-adjectives usually can't be used as nouns. The difference in usage is one of the keys to recognize which is which in Japanese grammar. For instance, 静か, which is the stem of a typical na-adjective 静かだ, CAN'T be a noun. We can't make a natural sentence using 静か as a subject or object, i.e., we can't say 静かがある, 静かを探す. We have to say 静かさがある, 静かさを探す instead, with using the noun form of adjectives. We can't use 静かの. 静かに acts only as an adverb "calmly/quietly/softly", etc, etc.
On the other hand, 健康 can be used both as a noun and a na-adjective. We can use 健康な人, 健康のために, 健康が一番いい, 健康を求める, 健康に暮らす, 健康にいい, and so on. In 健康に暮らす(to live healthy), 健康に is used adverbially as same as 静かに暮らす(to live calmly). 健康に can be considered a conjugation form of na-adjectives in this case. On the other hand, in 健康にいい(to be good for health), 健康 is used as a noun as same as 体にいい(to be good for body). に is the particle to indicate the target of いい here, not an infrection suffix of na-adjective.
This type of nouns/adjectives changes the meaning depending on how it's used, as a noun or as an adjective. One of the most well-known examples is "自由
な女神 vs. 自由
の女神". The former means "a free goddess/a goddess who behaves freely", whereas the latter is "a goddess of freedom/liberty", and it usually refers to the Statue of Liberty in New York. A famous linguist 奥津敬一郎, who named 僕はうなぎだ as うなぎ文, said that 自由
の女神は極めて不自由
な女神だ, since she has been standing at the end of Manhattan island for over 200 years without moving anywhere.
(Probably this can be an example of what Glenn-san mentioned.
)
The no-adjectives we are talking about here(e.g. 大荒れ, 上々, がらあき) have completely different characteristics from 静か, 健康 or 自由. They can't take が or を, so it's not a noun. They also can't take な when modifying nouns, so it's not a na-adjective. They basically act as an adjective, but take の when modifying nouns. That's why they are called no-adjectives. They are a bit special adjectives.