Nap and 40 Winks, at least, are interchangeable synonyms, both meaning a short period of sleep, with a special emphasis on those periods occurring during the day, or at least at a time other than when one is supposed to be sleeping (or turning in for the night, for the cultural meaning of 'the daily period of sleep at the end of the day').
So one could conceivably nap the entire afternoon away for a period of 4 hours, but at the same time one could turn in for the night at 2 am and get up and 6, and we wouldn't call it a Nap.
Definitions for both from the Merriam-Webster:
Nap: to sleep briefly especially during the day
Forty Winks: a short sleep; nap.
A Power nap is an expression coined by Cornell University social psychologist James Maas. It refers specifically to a short sleep of 20 (or sometimes 30) minutes or less which ends before the occurrence of deep or 'slow-wave' sleep.
Snooze is fairly interchangeable with the above words, but it introduces an element of lightness to the rest.
Snooze: to sleep lightly especially for a short period of time.
Kip seems to be a chiefly British word that's a lot more versatile. It may be the 'Britishness' of it that makes it feel 'old-fashioned' to non-British speakers, perhaps. And while it can be used to mean a nap, it can also refer to nightly turn in.
Kip: bed; eg. Get ready for the kip. OR
sleep, nap; eg. 1. Can I kip here tonight? and 2. After a rigorous walk, I needed to kip down a bit on the daybed.
As can be seen from the examples, kip is more interchangeable with sleep than nap. One can have a short kip or kip down for the night.
Siesta is a lot more specialized, though. One could siesta in winter, but you couldn't at 10 am in the morning or 5 pm in the evening. It specifically refers to a period of sleep in the afternoon, almost always post-lunch.
The special reference to certain countries and climates comes because in many of those countries, those post-lunch hours are (/were pre-air conditioning) too hot to do anything. So siestas are a culturally significant period of time, during which shops might be closed and schools might get out early during 'siesta time' in the summers.
Siesta: a regular period of sleep or rest in the afternoon in some hot countries; an afternoon nap or rest
Dates regarding etymology:
Nap: Possibly the oldest one here. From Old English hnappian "to doze, sleep lightly". In use from c. 1300, and in the construction "take [] nap" c. 1400.
40 Winks: Dr. Kitchiner, The Art of Invigorating and Prolonging Life (1821) Link
Power Nap: Maas, James B.; Wherry, Megan L. (1998). Miracle Sleep Cure: The Key to a Long Life of Peak Performance Link
Snooze: First used in 1789. The meaning "a short nap" is from 1793. Etymology seems to onomatopoeia referring to a snore.
Kip: Seems to have originated in around 1760s from the Danish word kippe (a hut or a mean alehouse) -> Irish slang term for a brothel (Earliest example in Oliver Goldsmith’s The Vicar of Wakefield (1761)) -> British slang for common lodging-house for tramps and the homeless (c. late 19th century) -> to the act of sleeping itself. The the modern informal or colloquial usage seems to have started in the twentieth century itself. Link
Siesta: The second oldest English word from 1650s, borrowed from the Spanish word siesta, from Latin sexta (hora) "sixth (hour); But its etymology might be the earliest thanks to the Latin root.
Best Answer
I think you're strongly overestimating how old that phrase is, if you think it is explained by ancient time-keeping standards.
The real answer is quite simple, and I'll quote from the OED:
And as you said, 'wee' is used as 'small'. Both idioms have recorded use in the 18th century, according the OED.