Which phrase attributed to the “cat-o'-nine tails” is the most credible?

In cat-o'-nine-tails, the cat was a stick and the nine tails referred to the nine lashes that were attached to it:

To show this use of “cat”:

OED at Cat (n)

†10 c. The cat-stick. Obsolete.

1636 Divine Trag. lately Acted 23 Sundry youths playing at Catt on the Lords day, two of them fell out, and the one hitting the other under the eare with his catt, he therwith fell downe for dead.

And to explain "playing at Catt"

Tip-cat

https://www.britannica.com/topic/tip-cat Tip-cat, also called One-a-cat, outdoor game dating back at least to the 17th century and introduced to North America and elsewhere by English colonists. The game was widely popular in 19th-century Great Britain and in early 20th-century North America.

Although there are many varieties of the game, all involve a stick about 3 ft (1 m) long, used as a bat, and a piece of wood (the cat) about 4 in. (10 cm) long, 1 to 2 in. (2.5 to 5 cm) thick, and tapered at the ends.

And thus

cat-o'-nine-tails n.

1. A whip with nine knotted lashes; till 1881 an authorized instrument of punishment in the British navy and army.

1695 W. Congreve Love for Love iii. i. 44 If you shou'd give such Language at Sea, you'd have a Cat o' Nine-tails laid cross your Shoulders.

It is clear from Congreve’s use that he and his readers/audience were familiar with the instrument. We can conclude that the name was used earlier, and indeed we have:

“Not enough room to swing a cat”

Swing (v.)

7. a. transitive. To cause to oscillate, as a body suspended from a support above; to move or sway (something) to and fro in this or a similar manner. to swing a cat (i.e. holding it by the tail); in no room to swing a cat in and similar expressions, said of a confined or narrow space. to swing the lead.

1665 Medela Pestil. 57 They had not space enough (according to the vulgar saying) to swing a Cat in.

I think the OED’s idea of swinging an actual cat to and fro is a little fanciful, and “cat” was, in fact a cat-stick, or any stick and to swing would mean to whirl about or wield a stick.

“Letting the cat out of the bag” There is no reason at all why a cat-o'-nine-tails should be kept in a bag – what would be the reason?

[edit 20201007 Phrases.org reports

"It is reported that the lashes were sometimes stored in bags, but the suggested nautical punishment origin fails at the critical point, in that it doesn't match the 'disclose a secret' meaning of the phrase."

Also - who reported it, and when and why is not mentioned: if it was to support the "cat out of the bag" theory then it should not be trusted.

More helpfully, it goes on to add

Of the two explanations, the 'pig in a poke' derivation is the more plausible, although I can find no direct documentary evidence to link 'letting the cat out of the bag' to the selling of livestock. Versions of the phrase exist in both Dutch - 'Een kat in de zak kopen' and in German - 'Die Katze im Sack kaufen'. These both translate loosely as 'to buy a cat in a bag', that is, to buy false goods.]

Thus the following may not be reliable

This is more prosaic. I don’t know whether you have ever tried to put a cat into a bag, but it is not an easy job as the cat resists, and once out, it runs off at high speed. Thus the secret figuratively escapes and cannot be brought back.

When was Cat-o-Nine-Tails first used?

The earliest instance for a proximity search of cat, nine, and tails in Early English Books Online is in The English rogue described, in the life of Meriton Latroon, a witty extravagant by Richard Head, published in 1668.

The last [dream] was of the Cat of Nine Tails, which my Master laid so home me thought that the smart thereof made me cry out, and so I awaked; (p.33-34)

Given that the term is used here without explanation, I find it plausible that the term was in use already by this time.

Not enough room to swing a cat

I found two earlier references to this expression.

The first is in The Lawes Resolution of Womens Rights by Thomas Edgar from 1632. Edgar describes a situation where a woman might be given the equivalent of a third part of her husband's chief house, and how that might not be a lot of room:

But how if the Common and all things bee so inclosed that there is not roome to swing a Cat in, women are not put in Rogum [the grave] with their Husbands any where but in the Indies [...]

The second instance is in the satire The first part of the last wil & testament of Philip, Earle of Pembrooke and Montgomery, Lord of Saint Quintin, &c., now Knight of Berk-shire from 1649. The main character is a fool giving his will and rambling on, which makes for a lot of wordplay:

I'de ra∣ther go to hel, or purgatory, where there's room enough for a coach and six horses to turn, then be mew'd up under ground, where a man cannot swing and [sic] Cat ; but prethee hold thy tongue, and let's mind our businesse, for I shan't die in quiet, til I see my dogs provided for.

I spot two related puns: mew as a verb could refer to cat noises ("mew, v.1") or to being enclosed ("mew, v.3"); the cat could be a literal cat or a stick ("cat, n.1," defs. 1 and 10). The proximity of cat and caring for dogs is also a possible point.

In both usages, I can't parse whether they mean a feline cat or some kind of stick. That could be part of the joke. I certainly see no evidence of a naval inspiration for this statement in either example.

That's interesting, but I thought the phrase "not enough room to swing a cat" came from the English folk game Dog and Cat, a pre-1600 game that led to the later development of Cricket and Baseball. The object of Dog and Cat was for one player to try to get a stick or a chunk of wood (the cat) into a hole that was defended by another player, also with a stick (the dog.)

https://en.wikipedia.org/wiki/Origins_of_baseball#Dog_and_cat