श्री गणेशाय नमः
The problem here is that parts of speech (adverb, conjunction, pronoun, etc.) are being used as labels to classify words as "being" one and only one part of speech, and that this categorization seems to be directed by faulty definitions.
Parts of speech are uses of words, and in English almost any word can be used in a number of such ways. Definitions of such use categories are just summaries, and not directions to be followed.
So, first of all, it is entirely possible for any English word to "be" more than one part of speech, depending on the context that word appears in.
In this case, the wh-words that have adverbial meanings
(temporal: when; locative: where; purpose/cause: why; manner/means: how)
are indeed used to introduce clauses, like many adverbs; but this doesn't make them conjunctions.
Conjunctions are one special class of words with special grammar.
When, where, why, and how belong, however, to a different special class of words, with equally special grammar. In fact, they constitute one subclass of this special class.
Most of the other wh-words (what, which, who) refer to nouns, so they're called pronouns --
either interrogative pronouns (because they're used to introduce question clauses)
or relative pronouns (because they're also used to introduce relative clauses).
Second, then, it's quite common for something other than a conjunction to "introduce clauses". Complementizers introduce complement clauses (including embedded wh-question complements), adverbs introduce adverbial clauses, etc. The traditional definitions don't go nearly far enough; they're rough guides, no more.
These are also the proper terms for the wh-words with adverbial senses. They came originally from Indo-European oblique noun cases like Locative and Instrumental, and they were used in much the same way as any other interrogative or relative pronoun, to introduce clauses.
Since they don't refer to nouns, however, clauses that they introduce have adverbial restrictions.
Relative clauses, for instance, must modify a noun, but if the relative pronoun is where, the noun it modifies has to be place, or a synonym, or some other word that refers to a place. Similarly, relatives headed by when must modify nouns referring to a point or length of time; the time when itself is most common.
Why and how are even more restricted.
How itself cannot be used at all as a relative pronoun, even if it's modifying the right noun:
- I don't like how he did that.
- I don't like the way he did that.
- *I don't like the way how he did that.
Why relative clauses must modify the noun reason. Synonyms don't work.
- I'm unclear on the reason why he did that.
- *I'm unclear on the purpose why he did that.
- *I'm unclear on the intention why he did that.
Essentially, the phrase the reason why has become frozen.
There's a lot more, but I forbear.