The reason for the western language learners' confusion when facing the so-called "two types of Japanese adjectives" is that they try to find similar constructs to their own native language in Japanese. And when they fail (since Japanese has no real adjectives at all), the naive learner or teacher (which unfortunately includes most textbook writers, who are not linguistically trained) will try to force their familiar concepts on Japanese.
What is an adjective?
Before we get to see what they "adjectives" in Japanese really are, we first have to define and understand the concept of adjective - otherwise it would be quite meaningless talking about them. I'll have to concede here that different linguistic schools have different ideas of adjectives, so the idea I represent here represents structural linguistics and especially the ideas of some of my own teachers. Some other schools (e.g. many functionalists) have not all too different ideas, but others, generative linguists in particular, may not agree.
An adjective is most simply a word class that is used specifically to qualify and describe nouns by being in agreement (i.e. matching gender, number and/or case) with them. A classical example would be the following Latin pairs, where as the noun changes, the adjective changes as well (note that word order is not important, and adjectives don't even have to be adjacent to their nouns):
Abacus (masculine) Terra (feminine)
Singular novus abacus nova terra New abacus/land
Plural novi abaci novae terrae New abacuses/lands
You've probably noticed that English adjectives don't really match this pattern, since they don't show neither gender, nor number nor case (they did show all of that in Old English, but not anymore). And indeed, the English adjectives are not quite adjectives in the classical sense, and the line is actually quite blurred between nouns and adjectives in English (Is 'winter' in "winter clothes" an adjective or a noun? And what about 'Hollywood' in "a typical Hollywood ending"?). In fact, English adjectives resemble the na-adjectives of Japanese more than Latin (and French, German, Spanish and Hebrew) adjectives.
Japanese counterparts of adjectives
Japanese has three (or more accurately two and a half) word classes whose most members translate to adjectives in English and other European languages, and therefore they are all usually called "adjectives" in the textbooks. These classes are
形容詞: Descriptive verbs
Descriptive verbs are also called stative verbs (verbs of state) or adjectival verbs by linguists, but most of us know these guys as i-adjectives or even true adjectives. Descriptive verbs are real verbs, although they miss some verbal forms that are available to other verbs. In essence, descriptive verbs are just verbs that come before the noun and form a relative clause. There's really no difference in syntax between 踊る蝶 (the butterfly who is dancing) to 白い蝶 (the butterfly who is white).
形容動詞: Descriptive nouns
Descriptive nouns are also called adjectival nouns by linguists, but most of us know them as na-adjectives or quasi-adjectives. Despite their name, they are no less adjectives than the "true adjectives", and some of them can actually be connected to nouns by using の rather than な. Historically, descriptive nouns were either nouns followed by the genitive の or nouns followed by な, which is the attributive (before-noun) form of the copula なり. So a の-adjective like
普通の男 started out as basically just saying "a man of ordinary", and
静かな場所 (which in classical Japanese meant:
静かである場所) started out as saying "a place which is silent".
It's instructive to note that descriptive nouns actually take a completely normal copula instead of な when past or negative forms are used (e.g.
静かじゃない場所), so these adjectives actually still use a copula to this very day, but it's most common form is replaced by the special な or の (depending on the adjective).
This class includes a rather small number of words that attach directly to the noun without a special copula and have no verb-like conjugation (like 形容詞). This includes words such as ある (some), いわゆる (so-called), 同じ (the same), たいした (great), この・その・あの, 大きな and 小さな. In fact, it's not really a real consistent class, but just a mish-mash of several frozen forms of words that used to be verbs (ある, いわゆる、たいした), descriptive nouns (大きな and 小さな), pronouns that merged with the particle の (この・その・あの) and the more special case of 同じ. There's nothing much to unite them, and this class is not productive (i.e. creating a new word in this class is very rare, and not at all trivial).
What's the state in other languages?
Well, the situation in Classical and Old Japanese was very much as it is today, except for descriptive nouns not using a special copula, but the regular copula instead. The conjugation of descriptive verbs was very different from today, but normal verbs also had a different conjugation.
I know Okinawan has at least descriptive verbs that are directly related to the descriptive verbs in Classical Japanese: e.g.
takasi (tall) from Old Japanese becomes
takasan. I guess it should also have a class of descriptive nouns, since they tend to be more productive than verbs, but I really don't know much about it.
As for Korean, it definitely resembles Japanese here. Descriptive verbs are conjugated like normal verbs but have some differences with them. Descriptive nouns have to be followed by the verb 하다 (hada - to do) instead of the copula, as we do in Japanese, but other than that they are very much alike.