Definitions of exists unique 
$\exists!x_0 \in S,P(x_0)$
Definitions:
1.$\exists x_0 \in S, P(x_0)\wedge(\forall x_1,x_2 \in S, P(x_1)\wedge P(x_2)\rightarrow x_1=x_2)$
2.$\exists x_0 \in S, P(x_0)\wedge (\forall x_1 \in S,P(x_1)\rightarrow x_0=x_1)$
$3. \exists x_0 \in S, \ \forall x_1 \in S, (P(x_1) \leftrightarrow x_0 = x_1))$


Question: I saw people sometimes use first one and sometimes use the second one in uniqueness proofs, are they equivalent, if so, is it possible to prove it?

Yes, they are equivalent. In fact, here is a third one that is equivalent:
$3. \exists x_0 \in S \ \forall x_1 \in S (P(x1) \leftrightarrow x_0 = x_1))$
To prove these three are all equivalent, let's show $1 \Rightarrow 2$, $2 \Rightarrow 3$, and $3 \Rightarrow 1$. I'll use the Fitch proof system. Please note that this system does not allow me to specify restricted domains, but that changes nothing:
First, $1 \Rightarrow 2$:

Then, $2 \Rightarrow 3$:

Finally, $3 \Rightarrow 1$:

