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I suspect she is a woman.

• Be able to live or at least spend long periods of time underwater in a lake

"Long periods" is relative and opinion based. Kimberly Jeffries can hold her breath for six minutes even after having drank a load of coffee. Also notice that women are easily capable of outsmarting men, specially men who do not pay attention to their surroundings. The lady of the lake may dive only when someone approaches, and then she can use a snorkel made of reeds to spend really long periods underwater.

• Preserve one or more European swords of at least arming sword length in good condition underwater for extended periods of time

Doable. I know plenty of women who have swords (I went to a fencing school). Preserving swords underwater would be hard but has been done. The sword of Goujian is approximately 2,500 years old and was found in a river without any tarnish. It is believed to have spent 2,000 years underwater.

• Possess one or more arms that can pass off as a human lady, preferably with the appearance of being clad in samite

A human lady clad in samite will perfectly mimic a human lady clad in samite. Also, women generally have woman-like arms^{[citation needed]}.

• From underwater, hear the sound of two empty coconut halves being banged together near the shoreline, so as to determine the Chosen One

As long as she is not deaf, that is a no-brainer. You can experiment yourself if you have access to a pool, a friend and a coconut.

Now let's see if I can collect a bonus.

• Actually looks like a human woman or mermaid, or at the very least has the silhouette of one while swimming when perceived from the shore.

You would be surprised to find that women look like women.

• Can talk to humans, so as to participate in farcical aquatic ceremonies that confer supreme executive power

For the vast majority of people throughout space and time, the first word they learned is "mommy" in whatever their native language is. This is usually taught by a woman, with most of our vocabulary being learned by imitating her. I think this is enough evidence that women can, in fact, not only talk to other fellow humans, but also teach them how to do so.

• Function normally on shore, also for the purposes of aforementioned ceremonies.

I am yet to see a woman fail to function normally on shore. Men, I've seen or read about plenty (Jackass and the Darwin Awards have a large set of examples of our shorely disfunctions), but women? Nope.

She is a normal woman standing in a submerged bin

While she awaits the chosen one she sits on the stool. As soon as she hears the coconuts, she stands on the stool and arranges her samite robe to conceal the edges of the bin. She raises the sword aloft and waits for the future king to appear.

Clearly there are variations. She could remain sitting with the samite already in place and simply raise her arm with the sword at the appropriate moment. She could have peepholes in the samite.

More complete diagram

Manatee are believed to have given origin to the myth of the sirens: their swollen breasts while breast feeding their younglings, their humanoid appearance, the human sounding sounds they can emit, together with the months long forced heterosexual abstinence experienced by old time sailors, could easily explain the "chanting naked ladies luring sailors to death with their voices".

Well, take a manatee adapted to lake or river life, and have it put out of the water a sword, which happened to be on the lake bed, right when a coconuts holder is passing by.

Some gentle moans of the manatee can sound to the ears of the adventurers like an invitation for a holy mission, of course said in a heavenly language not understandable by commoners.

Did you know some turtles can breathe through their butt?

Here we have a similar thing: a humanoid lady clad in long folds of her own skin, which just so happens to look like samite. Which, after some googling is apparently some kind of embroidered silk.

This species has a peculiar mating ritual: the males migrate over land towards ancestral spawning ponds and upon arriving there will tap together sticks (or similar objects, such as coconut halves) in a rhythmic fashion to entice the female to rise to the surface.

The female in turn presents the male with an object of choice. In a natural environment this is usually some kind of plant, shell, bones or even a dead mallard. In areas where humans are present the mating ritual will often involve either human detritus or even some votive offerings made by pagans in ages past.

Arthur is just quite blind, his servant very polite and the amphibian lady very desperate for a mate.

The poor man had terrible visibility and was a braggart.

No one was actually there. He came upon the remains of a sacked merchant cargo at twilight. The cargo, including thick samite tapestries, was dropped, clumped and draped over debris and branches near a bog or pond. It just so happened that the lumps looked like a reclining female form with an outstretched arm handling a sword.

Unfortunately, there were dead bodies around the booty. So creepy is as creepy does, and no one looked too closely at the reclining form. Imagine coming up to a place that smells of mud, rot, and offal, with the sounds of scavengers being heard throughout the woods. No one wants to linger there. On top of which, very few people would want to set foot on uncertain terrain when visibility is, at best, enough to see shapes and shadows.

If I was in their shoes, I would take the closest bit of valuable pickings, which any bit of metal would be, at that time. I would then head straight for the closest village for heat, food and safety. And when pressed for how I got such a valuable item (the sword), I would say "someone gave it to me". The details would come later with every mead-induced retelling.

In the West, the King in Orange blusters and bloviates endlessly in
barely comprehensible gibberish.

I see what you did there.

Here's an idea: the Lady of the Lake was once the Orange King's rival to the the throne. She has been banished from the dominion, or has gone into hiding to avoid being cast into the dungeon by the King's overzealous constables. Thus, she stays in a wooded lake area adjecent to a rebel province, hunting wildlife, sometimes hiding in the water, under the lilies and foliage, to avoid being seen by the King's spies.

The Lady of the Lake was once the consort of a past mad king who was long-since deposed. Perhaps she is of royal blood; perhaps she knows dangerous secrets of the kingdom. She is sometimes rumored by her enemies to be a sort of demigod, cyborg, or alien hybrid. Perhaps she knows enchanted spells, or is in possession of a magical or scientific artifact she received as a gift during her time as Queen. Even the Orange King is suspicious that many of his own court are secretly loyal to her.

Suppose also that the Orange King was in fact a known imposter, one who claimed to be wealthy but in fact had lived most of his life on borrowed wealth. His army was indisputably smaller than the Lady's, but while her forces were scattered about the coastlines, his much weaker forces were concentrated into the middle of the country, allowing him to put local pressure on the antiquated priestly caste and forcing them to arrange a coronation ceremony in his honor.

Whenever a company of knights go out into the woods, they sound their trumpets as they approach the lake. When she hears them, and sees their banner, she silently raises her golden-hilted, gemmed sword above the surface, bearing the ancient national insignia. She thus proves, it is indeed I, the Lady. She communicates from within the water, reflecting light off of the sword as a sort of heliograph, keeping her a safe distance from the King's archers who patrol the frontier. The knights then share her received wisdom with their kin and allies, keeping hope alive.

At what point in Arthur's description does it say the Lady of the Lake lives in the lake? She could just be a woman who lives in the general area of the lake and exerts control over it. The "ceremony" in the description only says that she was in the water when she gave him the sword.

There once was an old hermit lady who lived in a little cabin on an island in a large and deep lake. Once while gathering herbs along a path in the forest around the lake she found a rather nice sword, perhaps unknowingly dropped by a passing traveler. Several weeks later she heard the sound of clapping coconuts from across the lake and knew that brave knights were nearby. The sort of knights who might own a sword like she had found. Her boat was in poor repair at the time so she swam across the lake to greet the brave knights but the sword was rather heavy and she struggled with it. When Arthur reached the shore she summoned her last bit of the strength to lift herself and the sword out of the water but was too out of breath to say anything. Arthur took the sword and was so enamored by it that as he clip-clopped away to his destiny he didn't notice the poor lady was dragging herself onto the shore after nearly drowning.