Explanation of definition of normalizing: 9.1.12 in Terese "Term Rewriting Systems"?
I think you have misunderstood the definitions:

TERESE (9.1.1):

A strategy $S$ for an ARS $\rightarrow$ is a sub-ARS $\rightarrow_S$ of $\rightarrow$ having the same objects and normal forms.

This means that


If $a\rightarrow_S b$, then $a\rightarrow b$ (sub-ARS condition)
If there exists $b$ such that $a\rightarrow b$, then there exists $b'$ such that $a\rightarrow_S b'$ (same normal forms).


The second condition can be read as: if $a$ is not a normal form, then there is some rewrite step from $a$ which conforms to strategy $S$.

TERESE (9.1.12):

If $A$ is a superset of the set of normal forms for $\rightarrow$, a strategy $S$ is $A$-normalizing if the restriction of $S$ to terms not in $A$ is normalizing.

This means that a strategy is $A$-normalizing if every reduction sequence for $\rightarrow_S$ either


Ends on a normal form
Ends in $A$.


Clearly if $A=NF$ then only case 1. is relevant. However if $A$ is taken to be terms in head-normal form, then case 2. may occur independently of case 1.