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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.