It's a matter of definition, and <strike> hearkens back</strike> is similar to the 'One drop Rule' that used to be well known in the US.  Basically, to be a pureblood, you must come from magical families on both sides, not just two magic-using parents.  Conversely, **any** muggle blood makes you 'half-blood' at best.

From 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."

None of the families are really pure-blooded, as Dumbledore points out in Beetle the Bard

> "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..."

Harry's Mother's family is not a recognized wizard family, so he has muggle-blood.  Any muggle-blood with wizard blood defines him as a Half-blood.

To go straight to the horse's mouth, here's JK Rowling on the subject:<br>
(Note: This is from the Harry Potter Wiki, and I haven't been able to substantiate it as an actual JKR quote yet.)

> 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.