É uma questão de definição, e é semelhante à 'One Drop Rule' que costumava ser bem conhecida nos EUA. Basicamente, para ser um puro-sangue, você deve vir de famílias mágicas de ambos os lados, não apenas de dois pais que usam magia. Por outro lado, qualquer sangue trouxa faz de você 'meio-sangue' na melhor das hipóteses.
De Pottermore:
"The term 'pure-blood' refers to a family or individual without Muggle (non-magic) blood. The concept is generally associated with Salazar Slytherin, one of the four founders of Hogwarts School of Witchcraft and Wizardry, whose aversion to teaching anybody of Muggle parentage eventually led to a breach with his three fellow founders, and his resignation from the school."
Nenhuma das famílias é realmente de sangue puro, como Dumbledore aponta em Beetle, o Bardo
"So-called pure-blood families maintain their alleged purity by disowning, banishing, or lying about Muggles or Muggle-borns on their family trees. Then they 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..."
A família da mãe de Harry não é uma família bruxa reconhecida, então ele tem sangue trouxa. Qualquer sangue trouxa com sangue de mago o define como meio-sangue.
Para ir direto para a boca do cavalo, aqui está JK Rowling sobre o assunto:
(Nota: Isto é da Wiki de Harry Potter, e eu ainda não consegui comprovar como uma verdadeira citação de JKR.)
The expressions ‘pure-blood’, ‘half-blood’, and ‘Muggle-born’ have been coined by people to whom these distinctions matter, and express their originators’ prejudice. 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 maternal 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...the Nazis used precisely the same warped logic as the Death Eaters. A single Jewish grandparent 'polluted' the blood, according to their propaganda.