What is the difference/relationship between a 2-cocycle and a central extension for a Lie algebra?

Question

My understanding of a 2-cocycle is the following.

1. Let $(G,\cdot)$ be a group.
2. Let $(A,+)$ be an Abelian group.
3. Let $h:G\to \text{Aut}(A)$ be a homomorphism.
4. A 2-cocycle is a map $\varphi:G \times G\to A$ that satisfies $$ h(g_1)(\varphi ( g_2 , g_3 ) ) + \varphi ( g_1 , g_2 \cdot g_3 ) = \varphi ( g_1 , g_2 ) + \varphi ( g_1 \cdot g_2 , g_3 ) \quad \forall\quad g_1,g_2,g_3\in G. $$

Given this, we can define an Abelian extension of the group $G$ to be the set of all elements of the form $(a , g)$ where $a\in A$ and $g\in G$ alongwith multiplication law \begin{equation} \begin{split} (a_1,g_1) \cdot (a_2,g_2) = ( a_1 + h(g_1)(a_2) + \varphi ( g_1 , g_2 ) , g_1 \cdot g_2 ) . \end{split} \end{equation}

My understanding of a central extension is the same as the group extension above, except that $h$ is now the trivial map, so that $h(g)(a) = a$ for all $g\in G$ and $a\in A$.

Is all of this correct?

EDIT: The 2-cocycle definition I basically found from Wikipedia. The definition for central extension is something I worked out myself. The only difference I managed to find between the two was that $h$ is trivial for the latter. All I want to know is whether this is the only difference or if there’s something more subtle that I am missing here. Note that this is not for any course so I have no systematic introduction to these topics. I am just gathering what I can through Google. A simple yes/no or reference to an appropriate resource would suffice.