What is the definition of "pure-blood"?

Question

Being a "pure-blood" is an important aspect in the Harry Potter series, at least in the mind of some that consider themselves pure-bloods.

In the question Is there any such thing as pure-blood wizards in present day?, there were some interesting answers.

• Bellatrix quoted a letter from Dumbledore to Lucius Malfoy

  “So-called pure-blood families maintain their alleged purity by disowning, banishing, or lying about Muggles or Muggle-borns on their family trees. They then attempt to foist their hypocrisy upon the rest of us by asking us to ban works dealing with the truths they deny. There is not a witch or wizard in existence whose blood has not mingled with that of Muggles, and I should therefore consider it both illogical and immoral to remove works dealing with the subject from our students’ store of knowledge.”
  _{^{- The Tales of Beedle the Bard}}

  Note that this is not from the main books.

• Alex quoted

  In Chamber of Secrets, Dumbledore states that the Weasleys are pure-bloods:

  “And imagine,” Dumbledore went on, “what might have happened then. ... The Weasleys are one of our most prominent pure-blood families. Imagine the effect on Arthur Weasley and his Muggle Protection Act, if his own daughter was discovered attacking and killing Muggle-borns. ... Very fortunate the diary was discovered, and Riddle’s memories wiped from it. Who knows what the consequences might have been otherwise. ...”

  In Order of the Phoenix Dumbledore refers to Neville as a pure-blood:

  “He chose the boy he thought most likely to be a danger to him,” said Dumbledore. “And notice this, Harry. He chose, not the pureblood (which, according to his creed, is the only kind of wizard worth being or knowing), but the half-blood, like himself.

  In Half-Blood Prince Dumbledore confirms that there are pure-bloods by mentioning the possibility that Sirius's house is enchanted so that it can only be owned by a pure-blood:

  While his will makes it perfectly plain that he wants you to have the house, it is nevertheless possible that some spell or enchantment has been set upon the place to ensure that it cannot be owned by anyone other than a pureblood.”

  Especially the last example shows that there must exist some precise enough definition of the concept "pure-blood" that it is possible to create spells or enchantments that can distinguish between pure-blood and not pure-blood and apply it to an abstract legal concept like ownership.

It seems that these statements, all attributed to Dumbledore, contradict themselves. Or there are different definitions of the word that are used in the quotes.

So what is the definition or the definitions of the term "pure-blood"?

Answer 1

A single Muggle grandparent makes you a half-blood

Much the system of "magical blood" that JKR used in the Harry Potter series was drawn in an analogy to the Nazi's ideas of blood purity, even more than she had expected:

Section: F.A.Q.

Why are some people in the wizarding world (e.g., Harry) called 'half-blood' even though both their parents were magical?

The expressions 'pure-blood', 'half-blood' and 'Muggle-born' have been coined by people to whom these distinctions matter, and express their originators' prejudices. As far as somebody like Lucius Malfoy is concerned, for instance, a Muggle-born is as 'bad' as a Muggle. Therefore Harry would be considered only 'half' wizard, because of his mother's grandparents.

If you think this is far-fetched, look at some of the real charts the Nazis used to show what constituted 'Aryan' or 'Jewish' blood. I saw one in the Holocaust Museum in Washington when I had already devised the 'pure-blood', 'half-blood' and 'Muggle-born' definitions, and was chilled to see that the Nazis used precisely the same warped logic as the Death Eaters. A single Jewish grandparent 'polluted' the blood, according to their propaganda.

—JKR, in the FAQ on her old website (emphasis mine)

Answer 2

Pure-bloods are actually wizards with very little Muggle ancestry.

The “purest” wizards are the ones who can prove they have no Muggle ancestry for the furthest back through the generations. The Black family can trace their ancestry back to the Middle Ages.

“The tapestry looked immensely old; it was faded and looked as though Doxys had gnawed it in places. Nevertheless, the golden thread with which it was embroidered still glinted brightly enough to show them a sprawling family tree dating back (as far as Harry could tell) to the Middle Ages. Large words at the very top of the tapestry read

The Noble and Most Ancient House of Black
‘Toujours pur”
_{^{- Harry Potter and the Order of the Phoenix, Chapter 6 (The Noble and Most Ancient House of Black)}}

Ernie says he’s pure-blood since he can trace his ancestry back through nine generations of witches and wizards. This might indicate that nine generations of having no Muggle ancestry is enough to be considered “sufficiently pure-blood”.

“It was a very near miss,’ said Ernie. ‘And in case you’re getting ideas,’ he added hastily, ‘I might tell you that you can trace my family back through nine generations of witches and warlocks and my blood’s as pure as anyone’s, so –”
_{^{- Harry Potter and the Chamber of Secrets, Chapter 11 (The Duelling Club)}}

However, this leaves the possibility that some of his ancestors from ten or more generations back could be Muggle.

Pure-bloods often claim to be wizards with no Muggle ancestry.

To the strictest believers in it, being pure-blood supposedly means that the wizard has no Muggle ancestry at all. Other wizards are somewhat more lenient with what classifies someone as a pure-blood, so the definition may vary somewhat depending on who’s using the term. However, many pure-bloods claim that they have no Muggle ancestry at all, which is what Dumbledore refutes in his letter to Lucius Malfoy - Dumbledore’s saying even the purest wizard will have a Muggle ancestor. J.K. Rowling also stated this in the F.A.Q. of her old website.

Are all the pure-blood families going to die out? (We've lost the Blacks and the Crouches during the series)

Don't forget that, as Sirius revealed in 'Order of the Phoenix', none of these families is really 'pure' – in other words, they merely cross Muggles and Squibs off the family tree and pretend that they didn't exist. But yes, the number of families claiming to be pure is diminishing. By refusing to marry Muggles or Muggle-borns, they are finding it increasingly difficult to perpetuate themselves. This subject is touched upon in 'Half-Blood Prince'.
_{^{- Are all the pure-blood families going to die out? (F.A.Q. on J.K. Rowling’s old website)}}

Answer 3

Depending what kind of source you're looking for, Pottermore defines pure-blood like so:

The term ‘pure-blood’ refers to a family or individual without Muggle (non-magic) blood.

(https://www.pottermore.com/writing-by-jk-rowling/pure-blood)

This is the most precise definition I've found, however again it's not from a book or direct Rowling quote.