Is a syllable defined phonetically or etymologically?

Not all English speakers pronounce all words with the same number of syllables.
However there are general agreements as to how many syllables any word has. This is certainly true in singing.

In traditional music, one syllable will be sung on one note. Of course, if there are more syllables than notes, one "cheats" and adds notes. More notes than syllables, then one can "cheat: by distorting the word to created new syllables. Many songs have an even syllable to note equation. The "cheating" usually happens when there is an attempt to fit too many or too few words to an established tune,
In the well known song "Mary Had a Little Lamb", there are exactly the same number of syllables as notes in the tune. I have never noted any problems with this song as to syllabification.
Often persons who are not native speakers of English and speak English with an "accent" exhibit no such "accent" when singing English songs. Native and non-native speakers follow the same syllables with the same stress dictated by the tune. If one wished to assure the convention as to number of syllables is followed, one could do worse than to practice singing single syllable words on a single note

Almost always, the number of syllables in an English word is determined by the number of sounds (vowels) in the word; not the number of written vowels, but the number of distinct vowel sounds.

lit-tle

The second vowel in "little" is not written. Tl could not be pronounced without the understood vowel.
Etymology would have little to do with syllabification, unless one were trying to pronounce a current English word in the manner of some previous era. That does not occur very often.
I'm not sure this is a very serious issue for the most part. Probably few care if a syllable is added or subtracted in speaking as long as the meaning of the word is clear. Unless one is singing.

Syllables are only related to speaking. A syllable is determined by a vowel pronunciation. In the U.S., secretary is a four syllable word. In the U.K., it is usually only three syllables. Writing does not determine syllables. One's pronunciation or the pronunciation indicators in a dictionay determine syllables. As to "hour", it is a diphthong in southern U.S. (two vowel sounds gliding together in one syllable), but is two distinct syllables in the U.S. North, like ow-wer". Same goes for fire and oil. Church is very clearly one syllable, since it has only one vowel pronounced, unless you know of a locale where it is pronounced "chur-chah".

Sounds made without any vowel, such as "Whewww!" for relief of danger, are not words and have no vowel sounds. Shouting "Arghh" like a pirate could be considered a word with one syllable.

Words in a song are not nearly as exact as one might think. "A-may-zee-ing-grace" is sung as five distinct tones, but if one spoke this phrase in conversation, it would be only four syllables. In "Angels We Have Heard on High", the word "Gloria" is sung with many syllables or tones. Spoken, it is only three... unless a dialect pronounces it as "glor-yah". Then the second syllable is a diphthong. Many languages have diphthongs. In Mandarin Chinese, there are dozens, such as "tian" for sky / heaven / day or "guo" for a melon-like fruit. If you pronounced tian as two syllables like "tee-ann", most Chinese would not understand you. Same for guo pronounced like "Goo-wah".