Accurate description of the difference between the centralizer and the normalizer of a set $S$ in a group $G$? I've been reading some responses to the common question "What is the difference between the centralizer and the normalizer" and while I know it would make more sense to see if my understanding of the difference is correct by commenting on one of those post I do not yet have enough points to comment. So here's my understanding of the difference.
Give a group $G$ and a set $S$ in $G$ the centralizer of $S$ in $G$ is all the elements $g \in G$ such that given $y \in S$ we have $gyg^{-1} = y$. Whereas the normalizer of $S$ in $G$ is all the elements $g \in G$ such that for any $x \in S$ we have $gxg^{-1} = z$ for some $z \in S$. 
In other words it seems like a centralizer of $S$ in $G$ is the elements of $G$ which commute with all of $S$ and the noramlizer of $S$ in $G$ is all the elements $g$ of $G$ which sends $S$ back to $S$ when an element of $S$ is multiplied on the left by $g$ and on the right by $g^{-1}.$
Is that correct?
Do you have advice on better ways to understand the difference between a normalizer and a centralizer?
Is there a more concise way to describe the difference with words or set notation?
Is there an especially illustrative example to help one see the difference? (I have seen the example of $S_3$ and how $A_3$ is a centralizer but $S_3$ itself is the normalizer.)
 A: The answer already given is perfect. I just want to offer some alternative notation. One often sees the definitions
\begin{align}
N_G(S) &= \{g \in G \mid gSg^{-1} = S\} \\
C_G(S) &= \{g \in G \mid \forall s \in S, \ gsg^{-1} = s \}
\end{align}
In words, $N_G(S)$ is the set of elements of $G$ that fix $S$ under conjugation, and $C_G(S)$ is the set of elements of $G$ that fix each element of $S$ under conjugation. 
A: 
Is that correct?

Yes, you're correct.

Do you have advice on better ways to understand the difference between a normalizer and a centralizer?

"The only way to learn mathematics is to do mathematics." Paul R. Halmos.
Practice, basically.
That, and the fact that the normaliser $N_G(S)$ of $S$ in $G$ is the smallest normal subgroup of $G$ containing $S$, whereas the centraliser $C_G(S)$ is the group of elements that commute with all elements of $S$ (and no other elements).

Is there is a more concise way to describe the difference with words or set notation? 

Yes: use their definitions that use set building notation. See this other answer to your question, by @CharlesHudgins.

Is there an especially illustrative example to help one see the difference? (I have seen the example of $S_3$ and how $A_3$ is a centralizer but $S_3$ itself is the normalizer.)

What's wrong with the one you have already?
