This Wikipedia article talks about Lie algebra extension by a Lie-algebra, while this other artilce talks about extension by a module. This nLab article mensions central extensions by a ground field.
Are they all some special cases of something, or are they different concepts?
Certainly one can extend Lie algebras by Lie algebras. Here the ground field is considered as a $1$-dimensional abelian Lie algebra. For example, we have the short exact sequence of Lie algebras $$ 0 \rightarrow \mathfrak{sl}_n(K)\rightarrow \mathfrak{gl}_n(K)\rightarrow K \rightarrow 0, $$ where $\mathfrak{gl}_n(K)$ is an extension of $K$ by $\mathfrak{sl}_n(K)$.
In the second link, $M$ is a Lie algebra module, but it is in fact considered as an abelian Lie algebra in the short exact sequence. So this is not different.
The second Lie algebra cohomology $H^2(L,M)$ classifies equivalence classes of abelian extenions $$ 0\rightarrow M \rightarrow \mathfrak{g}\rightarrow L \rightarrow 0, $$ i.e., where $M$ is an abelian Lie algebra. See for example this post:
Classification of Lie Algebra extensions in Weibel's book
and other related posts.