[syndev]

But affine transformations are indeed not linear. The augmentation trick creates a new, linear transform in n+1 dimension, which is related but different to the affine transform in question.

The categorization in the article seems correct.

[dahart]

Yes, exactly, you're right. The article is correct, it's just not telling quite the whole story. Affine transforms aren't linear, and their augmented matrices that do the same thing are linear. In practice, it's both, and it's precisely cool because it's both.