Purity is slightly difficult to define here due to slight conflation of two meanings of the word.
As Cyberangel pointed out, Purity is directly related to the concept of
Kegare. Despite what one might think, it's not directly analogous to Sin, because Impurity does not necessarily pop-up due to conscious actions and sometimes it's due to natural events. Death is one example, people don't usually
choose to die, and others don't usually choose to be next to a dead person either.
Another hint of the nature of Purity is in Buddhism. One of the natures that can cause Impurity is
Yearning. Clinging onto the past or feeling discontent with your place.
Ghosts are also pure and so are celestials as well so should hermits have to purify themselves in order to ascend to Heaven or do their souls get purified once they get ascended? Are divine spirits and yaoyorozu no kami also pure since they're also spirits. Also why does being in a pure land make you not age? I think that there's a through line in this that would make everything else fall in place but I don't really understand it.
This is why Ghosts aren't Pure, they exist purely because they have a Fetter not letting them pass on. Hermits
should have to purify themselves to ascend to Heaven (Or Nirvana, if you want to use that word), Buddhists specifically seek to free themselves of all worldly desires (Yearning) to be Pure. This usually leads to a long life without short-cuts.
I say
Should because, as we all know, The Nai Clan became Divine Spirits after death and their servants (including Tenshi which makes her the techincal
third Miko) ascended to Heaven for their servitude during life. Is this nepotism? Yes. So it's entirely possible to gain Purity and access to Heaven by serving the Heavens and gaining their favour.
This also answers the question of "Are Divine Spirits and The Yaoyorozu no Kami Pure", yes they are. They gain such a state after ascending to Godhood. These aren't to be mistaken for the many Spirits we see in Ten Desires, as those are merely Greedy/Vulgar Spirits filled with Desires for Godhood.
The reason you don't age in a Pure Land is because Entropy, Decay, and Death is a source of impurity, and if you live in the Atmosphere of a Pure Land long enough you start to become more Pure and therefore more resistant to Death and Decay. The "Entropy" side is also what causes objects to not change their state much in a Pure Land. You can't really get rid of the Impurity in this way if you still live. The Lunarians still age, just very slowly, because they still have a small amount of impurity in them. The
Five Signs of The Dying Deva are also relevant since Constant Uneasiness (Yearning), and Withering of the Flower Crown (Death and Decay) are obvious signs of Impurity.
Apparently drinking the Hourai Elixir makes one impure because you are tainted with the desire for the removal of death but Mokou mentions says in IN: "To be devoid of death is to be devoid of life.
Without the unnecessary states of life and death,
I'm about as close as you can get to a pure human."
This is a bit tricky but the reason the Hourai Elixir makes you Impure is because drinking it forever marks you with the Desire (Yearning) to avoid death, therefore you are marked by death with the desire for the abscence of it.
Mokou is slightly correct when she says that she's about as close as you can get to a Pure Human. When you gain enough Purity you eventually stop counting as "Human" by turning into a Celestial, or Divine Spirit, or Hermit, etc. Mokou, curiously, still counts as a Human despite being Immortal. She doesn't age or truly die but she is rife with impurity, she has all of the hallmarks of Purity (unaging, deathlessness) but not the internal nature, she's inherently Impure from drinking the Elixir and, ironically, searching for a permanent way to die showing yearning for death, a source of impurity.
So to round this all up.
A "Pure" being would be someone who hasn't been exposed to Kegare (Impurity), has freed themselves from worldly desires (Yearning), and lives in a Pure Land (Heaven, The Netherworld, The Moon, a Senkai if they're a Hermit). They're
Amortal in the sense that Life and Death doesn't matter to their existence, they just
are, and therefore they don't age or even fall ill. By this definition every Celestial not named Tenshi Hinanawi is "Pure".
"Purity" is a state absent of Life, Death, Yearning, and Decay. Or rather, the abscence of "Kegare".
This is an unexpectedly complicated topic rooted in Theology, Mythology, and even weird Semantics. But I hope that this answer is understandable and satisfactory.
Now you know why In Defiance of Transience exists. I had to do some research myself for that.
