The 3rd grip is generally the correct grip for a chef's knife or santoku (the knife style used in the chapter).
The 1st grip is when you're using a cleaver to hack bones or maybe very rough, fast chopping (like large chunks of vegetables for stock). Then use the 3rd grip with a cleaver for for finer cutting/chopping, though there is a style of Japanese vegetable cutting (using a light, rectangular vegetable chopping knife) where 1st or 3rd grip are acceptable.
The 2nd grip just seems uncomfortable... there's less power behind the knife, index finger takes a lot of shock from all the chopping, palm and hand gets sore from a not very ergonomic grip.
Let's hope she makes a bento for our boy later on.