What's the difference between 静けさ and 静かさ?

Question

Much to my disbelief (no, I'm kidding, it happens all the time), I found out that my handbook was wrong to tell me we must use 静けさ, because 静かさ doesn't exist.

Wiki(tionary) says "For degree, corresponding to the English quietness, both 静かさ (shizukasa) and 静けさ (shizukesa) can be used, but 静けさ (shizukesa) may be more common." On the other hand someone on chiebukuro says:

「静かさ」といった場合「静かである程度」を 意味し、「静けさ」といった場合「静かであること（様子）」を意味しているように 思います。

meaning shizukasa is for degree, shizukesa is for appearance.

But I can't help feeling unconvinced. I've looked on weblio. In the ruigo jisho I've found 3 meaning for 静かさ (for reference 静けさ):

1. 音がないこと サイレンス ・ 物静かさ ・ 静寂さ
2. 静かな状況（例えば、話し声がしない状況） サイレンス ・ 音無 ・ 森閑 ・ 沈黙 ・ 無音 ・ だんまり ・ 黙 ・ 音無し ・ 黙り ・ 静寂 ・ 無言 ・ しじま ・ 無声 ・ 静けさ ・ 静寂さ
3. 穏やかな静寂 静寂 ・ しじま ・ 静けさ

So the first meaning, apparently, is the real difference with shizukesa, right?
But then again, if 1 is 音がないこと why 音無, 無音 and 音無し aren't also in the first group? And why 静けさ can be in the second group with these one, but not in the first group?
Side question: why 静寂さ for 1st and 2nd group, but just 静寂 in the 3rd?

For reference, instead of the 1st meaning of 静かさ, with 静けさ we have another group:
風のない穏やかさ 黙 ・ 沈静 ・ 無風 ・ 静寂 ・ 平静 ・ 静止 Isn't it overly specific? Can't this definition easily overlap with another group's definition (same of 2 and 3 for 静かさ).

Answer 1

Here is how I (and many other native speakers) would use the two words in real life. I am answering without looking at anything.

「静かさ」 describes the bare physical degree of how "not loud" a thing is. A high degree of quietness, while it may be desired, is not a prerequisite here.

Examples: 「静かさ」 is used to talk about how quiet a car, airconditioner, street, the way a person speaks, etc. is. Point is that those all make some noise if not a lot.

「静けさ」 means "tranquility". A high degree of quietness is an absolute prerequisite. One wants a certain quality in the quietness that is pleasing and satisfying to one.

Examples: 「静けさ」 is used to refer to the tranquility of a forest, mountain, lake, large park, etc. One can use the word to describe any natural scene the tranquility of which one appreciates.

I am not saying that everyone, myself included, uses these words correctly all the time, but this would be the general guideline that many would share deep inside.

Answer 2

I have found something that might be useful from poking around in the etymological information I have to hand.

Shogakukan notes in their entry for 静{しず}か that the noun form is 静かさ. There is no separate entry for 静かさ。 When looking to see if there was an entry for 静けさ, I found an entry instead for 静けし, which lists a noun form of 静けさ.

静{しず}けし appears to be an obsolete (or perhaps now dialectal?) form, derived as a 形容詞{けいようし} ("-i adjective") from the 形容{けいよう}動詞{どうし} ("-na adjective") that is 静{しず}か. The けし ending here is a suffix, for which Shogakukan provides the following definition (my additions in [square brackets]):

〔接尾〕体言や、形容動詞語根などに付いてク活用の形容詞を作り、その性質や状態を表わす。「露（つゆ）けし」「のどけし」「さやけし」「あきらけし」など。
(Suffix) Attaches to terms such as substantives [generally, nouns] or the stems of -na adjectives to form -ku-inflection adjectives [i.e. -i adjectives], expressing that kind of quality or state. [Examples include] tsuyukeshi "dewy", nodokeshi "calm, tranquil, serene", sayakeshi "crisp, clear, bright", akirakeshi "clear, distinct, clean", etc.

I suspect that this -keshi ending could be further decomposed into 気{け} "mood, spirit, sense, feeling" + the common adjectival ending し (classical terminal form for what became modern -i adjectives).

So ultimately 静かさ and 静けさ started out as the noun forms of separate words, with these then coming to be considered as alternate forms of a single word. I wonder if the separate etymologies might have some influence in the distinction described on Chiebukuro, where "shizukasa is for degree, shizukesa is for appearance."