I live in Liguria that is the home of Pesto and can confirm the recipe is correct. The traditional recipe also includes garlic, just be careful to not esagerate otherwise it will taste awful.
Usually it's either with nuts (so it's "Pesto alle Noci") OR with pine nuts (then it's "Pesto ai Pinoli").That is a fantastic endorsement! Do you generally add pine nuts or no? If I'm making pesto regularly, I add them, but this one was quite good (very herbaceous, which I liked) without the garlic and nuts, so I'll use it when I'm in a pinch, especially given that I grow an herb garden every summer, and basil grows like crazy in my backyard.