Orbit and Stabilizer Are the following definitions essentially the same:
Orbit:
Let $G$ be a group of permutations of a set $S$. For each $s \in S$, let $\operatorname{orb}_G(s)=
\{f(s) \mid f \in G\}$. The set $\operatorname{orb}_G(s)$ is a subset of $S$ called the orbit of $s$
under $G$
&
The orbit of a point $x \in X$ is the set of elements of $X$ to which $x$ can be moved by the elements of $G$. The orbit of $x$ is denoted by $Gx$:
 $Gx = \{ g.x \mid g \in G\}$.
I guess that they are essentially the same definitions. How?
 A: Let $X=S$, $x=s$ and $f=g$ and $\operatorname{orb} G(\cdot)=G\cdot$. Then the two definitions are the same.
A: The title seems a little different from the body. What you are describing in the body appears to just be two different notations describing the same set. But when you write this as the title

Orbit and stabilizer

it sounds like you are asking about the difference between the orbit and stabilizer.
The stabilizer of a point $x\in X$ is $\{g\in G\mid g\cdot x=x\}\subseteq G$.
The orbit of a point $x\in X$ is $\{g\cdot x\mid g\in G\}\subseteq X$.
The former one is "the actions that leave $x$ alone" and the latter is "all the places $x$ can go under the action of $G$.
A: Group ation is a homomorphism between a group $G$ and and the symmetric group of the $G-set$ $X$. If you know this, you can easily understand the first definition; that is, you see the elements of the group $G$ as the bijections on the set $X$. For the second definition, which is just the same,  you just skip the homomorphism part and consider the elements of $G$ which takes an element of $X$ to another one.
