Summary: Cagey may have originated in boxing in 1890s America, and used in a more familiar way the following decade.
Caged animals' behaviour
Owen Lattimore's 1990 China memoirs suggests:
"Cagey is American slang originated in the behavior of animals in a cage."
This is plausible; a caged animal can be wary, cautious and careful.
Dictionary review
After checking 33 dictionary definitions listed by OneLook, most say origin unknown and/or c. 1910 America, including Merriam-Webster, but claims "First Known Use: circa 1893".
However, Dictionary.com Unabridged ("Based on the Random House Dictionary") defines it as:
cautious, wary, or shrewd: a cagey reply to the probing question.
And also gives an earlier date range and explicitly gives the root as cage:
Origin:
1890–95, Americanism ; cage + -y
Additionally, a 1950 Slang To-day and Yesterday by Eric Partridge says:
Cagey. Cautious in giving confidence (-1900).
Cagey boxers in 1890s America
Here's some possible early uses.
From The Weekly Herald, August 21, 1892, in a preview of a night of boxing entitled "Scene And Personnel Of the Great Fights at New Orleans, September 5, 6, and 7.", written by Mike Donovan. The last fight of the night:
So while Skelly is a good man and clever enough, he has yet to get a good bit of professional experience before e can be expected to best Dixon. That man is bigboned, muscular, sinewy, nervy, "cagey;" his eye, his head and his hands work to gether all the lime. His body blows are terrors, and I must say that Skelly, fine as he looks, wil have a hard time of it against the clever Boston boy.
Later that year, cagey again shows up in a boxing context. The Morning Herald of November 1, 1892 has report on "Choynski The Victor: He Knocks Godfrey Out in Fifteen Rounds" at Coney Island on 31st October:
Round Eleven— From the cagey manner in which this round was there was every indication that it would prove a long battle. Both men had their left eyes almost closed, but with this exception they looked as though they could last for a week. All the punching that was done was light with the exception of a few stomach punches.
Next, the variant cagy shows up in another boxing report, published in both the Warsaw Daily Times and Aurora Daily Express (and probably the Atlanta Constitution) of March 9, 1893:
Round 1.-Both men came to the scratch with a jauntiness which showed their seeming lack of care. They commenced work in a very cagy manner until Fitzsimmons finally led with the left and fell short of his mark, Hall's left on the counter barely touching his face.
1900s American newspapers
The following decade we can find more familiar uses of the term. I've noted which of these are pay-per-view as I've not paid to verify the OCR is correct.
Pay-Per-View - The Sun - Apr 6, 1900
There is something gallant, says. the Brooklyn Eagle, even if amateur, in the Admiral's candid, brief, albeit somewhat cagy. announcement.
A story on a "Magical Wine Cask" in the Lawrence Daily World of Oct 2, 1901 uses cagey twice to mean careful:
The labels read "Assmannshauser." There was not a suggestion anywhere of "Oberingelheimer." The dealer had remarked with a sly wink: "I am very cagey on wines; nobody can beat me in the selection". But the box was nailed up, and a note informed the seller that a mistake has been made. Hearing nothing in reply the purchaser called for an explanation, says the New York Press.
"So they sent you Assmannshauser instead Oberingelheimer, did they?" said the cagey man."Did you taste it? No? If you had tasted it you would have seen at once it was Oberingelheimer. My man made a mistake in pasting the wrong labels. I will send for the case and have the labels changed, if you like, but it will make no difference in the taste of the wine, because A and O are drawn from the same cask. These are the only two brands of red Rhine wines, and only the shrewdest expert can tell which is which."
Pay-Per-View - Detroit Free Press - Jul 6, 1902:
Mercer pitched his usual cagey game, and the fourth, mentioned above, was the only one in which more than one hit was registered against him.
Pay-Per-View - Detroit Free Press - Aug 28, 1909:
Pltle got cagey when askedi where he had taken the single load of fttrnititre and things. and decided not to tell. except that he had driven them to ...
The Evening Sentinel of Aug 19, 1910, in an article about former president Theodore Roosevelt titled "The Colonel Is Saying Nothing. Refuses To Discuss Breach Between Himself and Taft":
In case the old guard goes to Saratoga and tries to undo what it has already done, which is mighty doubtful, they might realize that with New York lost the colonel [Roosevelt] would be a good man to captain the sinking ship and go down with it. The colonel is pretty "cagey" when It comes to matters of this sort. For that reason he believes that he had bolter wait and make his onslaught of a ...
1900s American serial dime novels
Cagey is used in the modern sense in the New York nickel weekly Secret Service: Old and Young King Brady, Detectives.
The Bradys and Dr. Hop Low; or, The Deepest Mott Street Mystery of January 4, 1907 by A New-York City Detective:
"Strange."
"Very. The place seems to he hoodooed."
"Did you ever ask Hop Foon about it?"
"Yes."
"What does he say?"
"He can give no reason."
"Or will not. These Chinese are very cagey."
"Don't I know it! I ought to. I’ve been dealing with them for the last ten years."
The next year: The Bradys and Little Chin-Chin; or, Exposing an Opium Gang of January 31, 1908:
“Shall you arrest them?” demanded Charley.
“Not yet,” replied Old King Brady. “Of course, they must have heard of the notice on the bulletin, but they are too cagey to show their hands. No doubt they backed this man Ramage up with money to embark in the jewelry importing business. We will wait a bit. and see what move they make."
Footnote: 19th century Scottish cheerful
There's an earlier Scottish term cagey which may be unrelated, and means cheerful or joyful.
From the addenda of Two ancient Scottish poems: the Gaberlunzie-man, and Christ's Kirk on the Green by James V (King of Scotland), John Callander of 1782:
Ver. 8. Cadgily] The word cadge is probably derived from the Sclavonian chodge, to trudge on foot; whence, too, our so fentgy[?], a little wench, who does the dirty work in a farmer's kitchen. The word cadgy, in the present case, should, I think, be written cagy, or cagie, which would agree better with the pronounciation [sic]. It imports merry, chearful, jovial, and is, I believe, an abbreviation of the old French word cagedler, the same with cajoler, to cajole, slatter, cox.
There's also another note about Cadgily which links it to cadgers who carry goods for sale in cages, also called creels, "who use to sing, in order to beguile the tediousness of the way. From primitive ca, cad, cap, anything made for containing, possibly from Gaelic cadhla.
This Scottish word is used in Blackwood's Edinburgh magazine: Volume 9 of 1821:
It was such an early hour that the steward, not counting on any body wanting to breakfast till they would reach Greenock, had made no provision of provender; so that when I went to him, as cagy as a pyet [magpie] picking at a worm, to inquire when the eggs would be boiled, judge of my mortification to hear that there was to be no breakfasting that morning; which disappointment, with the natural vapours of the river's tide, caused me to remember the judicious observe of Mrs M'Leckit, that there was a danger in going on the water with an empty stomach. However I had put some ginge-bread nuts in my pocket, and by the use of them the wind was keepit off my heart, and I suffered less from the effect than might have been expected.
And in 1821's Annals of the parish: or, the chronicle of Dalmailing by John Galt:
... bed at Kilmarnock, where they stopped that night ; but when they came back to the lady's in the morning, she was as cagey and meikle taken up with them, as if they had gotten her full consent and privilege to marry from the first.
Meikle is a variant of mickle: chiefly Scottish meaning great, much.
And more from John Galt, 1834's The literary life, and miscellanies: in three volumes: Volume 3, has a cheerful meaning of cagy:
She just had some milk in a saucer, and, rolling up a bit of rag, she dipped it in, held it to the bairn's mouth, and he was as cagy about it as if it had been a pap of the motherly gender.
And again in the same volume:
But I had little though of that them, for it was a blithe [joyous] morning, with the dew on the grass, and the laverock in the lift, and I was as cagy with the thought of going to do for myself as if my een were haill, and I a skipping mawkin [hare/simpleton].
Footnote continued: early 20th century American sexual desire
A snippet from 1917's Dialect notes, Volume 4 by the American Dialect Society says:
cagey, cajy (k édzi), adj. Having strong sexual desire ; esp. of a male. “The stallion is quite cajy after seeing a mare go by.”
And a snippet from their 1928 Dialect notes, Volume 6 says:
“Cagey and horny are the ordinary words for sensual, and are never used in polite company.
This has roots from the Scottish cagey, joyful, as Michael Montgomery 2006's From Ulster to America: the Scotch-Irish heritage of American English explains:
cadgy, cagey, caigey, caigy adj Spirited, sportive (in the US also amorous, sexually aroused), [origin uncertain; OED cadgy adj 1 'sexually excited' cl724—>, Scottish and northern dialect, 2 ‘cheerful, merry, glad, 1725->; SND cadgy ‘cheerful, in good spirits; dotingly amorous; eagerly willing’; DARE cadgy adj ‘lively, excited, spry; especially of stallions: sexually aroused’ chiefly South Midland]
ULST:
1880 Patterson Antrim/Down Glossary 16 caigey = in very good spirits, lively, wanton, eager.
c1910 Byers Glossary, cagey = in good spirits, gay, cheerful, sportive: ‘He’s a cagey boy’.
1953 Traynor Donegal Glossary 42 cadgy = gay, cheerful, in good spirits.
2000 Fenton Hamley Tongue 31 caigy = full of sexual desire.
U.S.:
1917 Kephart Word-list 409 caigy = full of sexual desire.
1955 Parris Roaming Mts 177 The dogs are cagey. They are born to taunt rather than fight.
1993 Walker Life History 30 [A powder was] used to make the animals ‘cagey’ and reproduce.
Best Answer
Taken from Online Etymology Dictionary
lie (v.1)
lie (v.2)
You can see that both words have different roots (legan v. licgan) but converged together in terms of spelling. It is apparently, totally coincidental as you can see there is no similarity in meaning or prior spelling along the way.