Definition of nucleophile
All nucleophiles are Lewis bases. They are just electron pair donor.

A molecule or ion is called a "base" (or, “Brønsted base”) when it interact with proton, it is called a "nucleophile" when it interact with other atom centers, which can be carbon, nitrogen, sulfur and so on. So the row "base" and "nucleophile" may overlap. For example, $\ce{OH-}$ is a base when it take proton off an acid, and it can also be a nucleophile when it hydrolyzes ester (nucleophilic attack to carbonyl group).

The broad nucleophilicity definition is how good the nucleophile interact the positive or partial positive atom center. When the atom center is proton, nucleophilicity is then basicity. Because proton center is hard and usually less steric hindred, a bulky and hard (for hardness $O$ > $N$ > $S$ center) electron donor will be likely to prefer proton over the softer and more hindred non-proton center. These electron donors are called Non-nucleophilic bases. Potassium tert-Butoxide is a good example.