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 [wizard] 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.
It was also Rowling's mouthpiece in the novels, Albus Dumbledore, who said:
You place too much importance, and you always have done, on the so-called purity of blood!
You fail to recognise that it matters not what someone is born, but what they grow to be!
Coincidentally (perhaps), a 2016 study linked reading Harry Potter to a lower opinion of Donald Trump.