In the Stacks Project, Tag 032F, we find:

Definition. Let $R$ be a domain with field of fractions $K$.

1. We say $R$ is N-1 if the integral closure of $R$ in $K$ is a finite $R$-module.
2. We say $R$ is N-2 or Japanese if for any finite extension $L/K$ of fields the integral closure of $R$ in $L$ is finite over $R$.

Why do they call these rings 'N-1' and 'N-2'? What is the reason behind this terminology? (Other sources denote them the same way.)

(I guess the 'Japanese' terminology for the latter is to pay tribute to the Japanese mathematicians that studied these kinds of rings properties, such as Masayoshi Nagata.)