Why have Jack or John have been used as euphemisms to refer to a toilet in particular?

The article below describes the origin of the flush toilet by **John Harington and also references the Elizabethan slang for Toilet which was the Jakes It is not a great leap of imagination to link Jakes with Jack's, especially with Irish pronunciation where I believe it is the common term for the toilet History.com Therefore it looks like the terms John and Jacks are not derived from the same source albeit that Jack is a commonly used nickname for John

The first modern flushable toilet was described in 1596 by Sir John Harington, an English courtier and the godson of Queen Elizabeth I. Harington’s device called for a 2-foot-deep oval bowl waterproofed with pitch, resin and wax and fed by water from an upstairs cistern. Flushing Harington’s pot required 7.5 gallons of water—a veritable torrent in the era before indoor plumbing. Harington noted that when water was scarce, up to 20 people could use his commode between flushes. Harington described his device in a satirical pamphlet entitled ‘A New Discourse on a Stale Subject, called the Metamorphosis of Ajax”—a pun on the term “a jakes,” which was a popular slang term for toilets. Although Harington installed a working model for Queen Elizabeth at Richmond Palace, it took several centuries—and the Industrial Revolution’s improvements in manufacturing and waste disposal — for the flush toilet to catch on.

In 1775 English inventor Alexander Cumming was granted the first patent for a flush toilet. His greatest innovation was the S-shaped pipe below the bowl that used water to create a seal preventing sewer gas from entering through the toilet. In the late-19th century, a London plumbing impresario named Thomas Crapper manufactured one of the first widely successful lines of flush toilets. Crapper did not invent the toilet, but he did develop the ballcock, an improved tank-filling mechanism still used in toilets today.