As far as I can tell, Noether normalization uses the term "normalization" in the English sense, that something has been given a standard form. And as such it's not intrinsically related to normalization in the sense of "take the integral closure in the ring of fractions". I would leave the matter at that, except that in book after book, the two are introduced and studied in very close proximity. Am I missing something?

What is the statement of Noether normalization theorem you are referring to? The one I know is the following, that you can find for instance in Shafarevich's book "Basic Algebraic Geometry I", page 65.
That means that any integral domain $A$ which is finitely generated over the field $k$ is integral over a subring isomorphic to a polynomial ring. In particular, let $X \subset \mathbb{A}^{n+1}$ be any degree $d$ hypersurface. Then its "normal form" is given by a monic equation $$z_{n+1}^d + f_{d1}z_{n+1}^{d1}+ \ldots + f_1 z_{n+1}+f_0=0,$$ with $f_i \in k[z_1, \ldots, z_n]$, and the map $\varphi \colon X \longrightarrow \mathbb{A}^n$ in Noether theorem is simply the projection onto the first $n$ coordinates $z_1, \ldots, z_n$. This shows how the two concepts of "normalization" are closely related. 

