Since the terminology "normal", "normalized", etc has different meanings in mathematics (some geometric in flavor, like when referring to perpendicularity) and I just read in Eisenbud's book on Commutative Algebra that what I knew as the integral closure of an $R$-algebra $S$, where $S$ is a commutative ring, is also called the normalization of $R$ in $S$, then I cannot help but wonder where does this terminology come from?
In the introductory section of chapter 4 in Eisenbud's book, he gives some examples that ilustrate how the normalization of a ring behaves "better" than the ring itself. Some are related to algebraic geometry and algebraic varieties, and others to algebraic number theory and algebraic number fields.
So my question is the following.
What is the origin and what motivated the terminology of normalization in commutative algebra?
Thank you very much.