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.
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"?