Question: I assume that arawaru and arawaruru are simply more archaic ways of writing arawareru, presumably changing in the late 19th or early 20th century. But was/is there any significant difference/implication/contextual or formal reason for using arawaru (現る, 顕る, 表る, あらわる) or arawaruru (顕るゝ, あらわるる)?
Your question touches on a common phenomenon in Japanese: the shift from the older 下【しも】二段【にだん】活用【かつよう】 or "lower bigrade conjugation" pattern to the modern 下【しも】一段【いちだん】活用【かつよう】 or "lower monograde conjugation" pattern, also described as "type 2" in English-language materials. The "lower" part refers to the vowel -- "lower" verbs had stems ending in -e, and 上【かみ】 or "upper" verbs had stems ending in -i.
In a nutshell, all the modern type-2 verbs ending in -eru used to end in -u in the 終止形【しゅうしけい】 or "terminal form" used to end a sentence, or used as the headword in a dictionary. So modern 食【た】べる used to be 食【た】ぶ as the plain form, and 顕【あらわ】れる in your example used to be 顕【あらわ】る.
But that's just for the 終止形【しゅうしけい】. The 連体形【れんたいけい】 or "attributive form" is used when the verb modifies a noun. In modern Japanese, the 終止形【しゅうしけい】 and the 連体形【れんたいけい】 are the same thing, but in Classical Japanese and older stages of the language, the 連体形【れんたいけい】 was a separate conjugation, formed by adding a る on the end of the 終止形【しゅうしけい】. So for 食【た】ぶ, the attributive was 食【た】ぶる, and for 顕【あらわ】る, the attributive was 顕【あらわ】るる. In your sample text above, 顕【あらわ】る is used to modify 圖【ず】, so the verb is conjugated in the attributive, as 顕【あらわ】るる.
Over time, the conjugation shifted, and various forms fused, so the -e stem ending came to be used as the basis for all forms: giving rise to modern stem forms 食【た】べ and 現【あらわ】れ, and fused terminal / attributive forms 食【た】べる and 現【あらわ】れる.
For more on the conjugation pattern, see the Japanese Wikipedia article. The box at the upper right of that page has links to the other classical and modern conjugation patterns, each with tables showing the different forms -- 未然形【みぜんけい】 (incomplete), 連用形【れんようけい】 (continuative or stem), 終止形【しゅうしけい】 (terminal), etc.
Question: Why is 現われる or 顕われる read/voiced/understood as arawa-re-ru instead of arawa-wa-re-ru?
This has to do with spelling conventions, and some looseness over time in what sounds were regarded as 送【おく】り仮名【がな】, the kana that typically follow the kanji to show inflectionary endings.
The verb arawareru, as shown above, comes from older form arawaru. The conjugating portion of the older form is clearly just the -ru on the end, which becomes -reru in various conjugations, and the modern 送【おく】り仮名【がな】 convention is to spell the term as 現れる, with the -wa- internal to the kanji and the classically conjugating -reru spelled out explicitly (even though, in modern Japanese, again only the -ru on the end changes at all; including the れ in the kana might help avoid ambiguity). However, this is only a convention, and not carved in stone, and older spelling conventions usually spell out the -wa-, as 現われる. My copies of Daijirin and the Kokugo Dai Jiten both list both spellings, with the -wa- either explicitly spelled out in kana, or left implicit.
Consequently, both 現れる and 現われる are considered to be valid spellings of arawareru. Don't add an extra -wa-. :)
A little speculation:
Monolingual dictionaries show that the older pre-20th-century-reform spelling was あらはる. If you dig around in etymologies enough, you'll notice that the はる ending here sometimes suggests a derivation from an older verb form ending in ふ. We find that あらふ is the pre-reform kana spelling of modern 洗【あら】う, and sure enough, one of the less-common senses of this verb is "to bring something hidden into clear view". The passive conjugation of 洗【あら】う is 洗【あら】われる, and the meaning is "to be brought into clear view [as of something hidden". This is quite close to 現【あらわ】れる, "to become evident, to be discovered". I suspect that 洗【あら】う is the root of 現【あらわ】れる, and as such, the -wa- would actually be part of the conjugable portion -- which might explain why older spellings explicitly spell out the わ in kana.
But again, this last part is only my own speculation, and is not listed as the derivation in any of the dictionaries I have to hand.