I would say a good rule of thumb is to look at the base form of the word.
If the base form of the word ends in a "y," then a noun form ending in "-ian" will generally be pronounced as two syllables. Intuitively, these words have a hard time losing that "ee" sound. For example:
- comedy => comedian
- custody => custodian
- history => historian
- library => librarian
- ovary => ovarian
If the base form of the word ends in a "c" or "g" (velar stops), then that originally hard "k" or "g" sound becomes "sh" or "j" respectively in front of the front vowel "i" of "-ian" and is pronounced as a single syllable (a form of elision). A similar transformation can happen for words that already end in "s" or "sh" sounds. For example:
- clinic => clinician
- cosmetic => cosmetician
- magic => magician
- theology => theologian
- Venice => Venetian
If the base form ends in most other consonants or vowels other than "y", then the "-ian" ending cannot be elided with the final consonant/vowel into a single syllable, and thus is pronounced as two syllables. In the case of final vowels, an infix "-n-" is generally used to separate the final vowel from the "-ian" ending. For example:
- guard => guardian
- pluto => plutonian
Finally, examples like "Asian" are in fact a slightly different construct, wherein the suffix "-n" is used to make an adjective. Words that would be confusing in this regard because they end in "-ian" include:
- Asia => Asian [one-syllable "-ian"]
- mammal => mammalian [two-syllable "-ian"] (here the root is really the Latin "mammalia" which is the term used for the Class of mammals)
The website you linked is unreliable and incorrect. I wouldn't recommend using it (or any other website) syllabification because syllabification is a highly controversial topic in linguistics. How to syllabify a word depends on who pronounces it. For example, the traditional pronunciation of the word film is [fɪlm] (one syllable), but I've certainly heard the disyllabic pronunciation [ˈfɪ.ləm].
The word 'dune' is usually pronounced [djuːn] and 'rider' [ˈɹaɪdə] (BrE). I agree with your analysis that 'dunerider' should be [ˈdjuːn.ɹaɪ.də] (three syllables) (it can also be pronounced in many other ways, though, even four-syllable pronunciation is possible depending on the speaker). It is not pronounced /ˈd(j)uː-n-raɪ-də/ (four syllables), because it suggests that the /n/ is syllabic which it is not. /n/ can only be syllabic when it's preceded by an obstruent (/t s z d/ etc) as in button. I would suggest you use your ears for deciding the number of syllables in a word.
Let's see how 'dunerider' is a three-syllable word:
This is the sonority curve for the word 'dunerider'. The dark blue dots mark the peaks of sonority and the number of peaks correspond to the number of syllables. As you can see, there are three peaks in [ˈdjuːn.ɹaɪ.də], hence three syllables.
The sonority of all phonemes of English can be depicted on a sonority scale (sonority hierarchy). A sonority hierarchy is a hierarchical ranking of speech sounds. Typical order of sonority values is:
Vowels [ɑ, ɔ, ɪ, i] etc > Glides [j, w] > Liquids [ɹ, l] > Nasals [m, n, ŋ] > Fricatives [s, f, θ, ð, z, ʃ] etc > Affricates [d͡ʒ, t͡ʃ]> Plosives [p, b, t, d, k, g]
Vowels are the most sonorous whereas plosives are the least sonorous sounds.
[From my previous answer]
I have explained the Onset Maximisation Principle (syllabification rule) in this answer.
Best Answer
Pluralisation never changes the stress in English words; plurals are stressed on the same syllable as the singular. Begonia is stressed on the second syllable—/bɪˈɡəʊ.nɪə/—and so is its plural.
Moreover, the plural marker -s/es doesn't add an extra syllable to a word unless it ends with /s/, /z/, /ʃ/, /ʒ/, /t͡ʃ/ or /d͡ʒ/.
In case of begonias, the final sound (the sound before the plural marker) is a vowel so it's a three syllable word.