Prone vs. Supine vs. Prostrate
If you didn't already know this distinction, you can sometimes infer it from clues in the context:
Just now, she chose to drink from the creek, lying prone on the ground, her face half-buried in the water, and this, not because she was thirsty, but because it was a new way to drink.
—Frank Norris, The Octopus: A Story of California, 1901
You could visit the aft cabin by lying supine on a wheeled cart and pulling yourself along an overhead rope through a tunnel 85 feet long and two feet in diameter.
—Daniel Ford, Air & Space, April/May 1996
In some instances, even without contextual clues, one can make an educated guess whether the person is face up or face down:
Foxy, in a skirted lemon-yellow maternity swimsuit, lay supine on a smooth rock, eyes shut, smiling.
—John Updike, Couples, 1968
Her body had to be positioned prone on the operating table, her forehead resting in a sling, her shoulders held by crutchlike supports.
—Barbara Harris, Cosmopolitan, August 1972
Other times, it might not be so apparent:
I too have been prone on my couch this week, a victim of the common cold
—Flannery O'Connor, letter, 20 Mar. 1961
And in yet other times, prone is used to describe one who is quite clearly facing upward:
He lies prone, his face to the sky, his hat rolling to the wall.
—James Joyce, Ulysses, 1922
Both prone and supine also have meanings that have nothing to do with physical position. Supine, with its image of one lying comfortably idle, shows use in the sense of "mentally or morally slack" or "permissively inactive":
Some have called him Florida's first king, though that was before budgets tightened and the legislature, formerly as supine as Huey Long's once was in Louisiana, began showing signs of independence and even resentment toward him.
—David Margolick, Vanity Fair, July 2001
He despised the farm lobby, for example, for its mastery over a supine Congress that in the early days of the past century regularly enacted costly favors for farmers while extolling special virtues it attributed to tillers of the soil ...
—Russell Baker, The New York Review of Books, 11 Nov. 2010
Prone is used in the sense of "having a tendency or inclination," as in "prone to worry" or "accident-prone." This usage is similar to such words as apt, liable, or likely (as in "apt to be late"), but in many instances prone implies a vulnerability to attack or damaging influence, much like one who is lying face down and cannot see what is approaching:
Take coronaviruses, for instance. While all kinds of animals play host to them, bats seem especially prone.
—Carrie Arnold, New Scientist, 8 – 14 Feb. 2014
With all the rain we've had recently, my roof is beginning to leak and my basement has become prone to flooding.
—Marc Fitten, The New York Times, 21 Feb. 2010
And then there is prostrate, which means "stretched out with face on the ground in adoration or submission" or "completely overcome and lacking vitality, will, or power to rise."
Added to the mix are the cameras that broadcast the daily Mass (that, pace Mahony, is said in Latin and accompanied by nuns fully prostrate behind the grille but not beyond the cameras, which are mounted on the walls like a series of surreal, postmodern Stations of the Cross).
—J. V. Long, Commonweal, 14 Aug. 1998
Ordinary Munchausen syndrome (a term coined in 1951) is characterized by a morbid desire for medical attention: its victims induce symptoms in themselves for the thrill of arriving prostrate in the emergency room and being swarmed over by doctors and nurses shouting complicated instructions.
—Larissa MacFarquhar, The New Yorker, 9 Mar. 1988
By 1978 John Lennon had ceased to resemble himself.... Totally prostrate, John couldn't even raise his head to look at the TV screen; instead, he employed a pair of trick glasses with prism lenses that enabled him to watch while lying flat on his back.
—People, 22 Aug. 1988
Angola, whose economy is prostrate after a decade of civil war and foreign invasions, depends on oil and oil products for about $2 billion--or 809 percent--of its foreign exchange earnings.
—James Brooke, The New York Times, 5 Jan. 1985
So while prone, supine, and prostrate have specific meanings with regard to position, they also come with situational connotations in many cases: prone suggests exposure or vulnerability; supine connotes a position of willful inactivity; and prostrate often describes a giving in to forces of lethargy or submission.
In summary: A person lying prone is facing downward. A person lying supine is face up. Prostrate can be applied to someone either face up or face down: a prostrate person is either stretched out with their face on the ground in adoration or submission, or simply lying flat.