Grasping the idea of Virasoro Algebras in 2D Conformal field theory

Question

I have been trying to understand the connection between Virasoro algebras and CFT. After a course in string theory, I was under the impression that the Virasoro algebra was simply the Lie algebra of the conformal group, but that is not the case. I would like to get an overview of what is going on, since right now I do not have time to go into the details, but would still like to have a bird's eye view of the topic.

This is what seems to me so far, trying to translate physicists' lingo to maths (in the context of CFT in the euclidean plane):

1. The conformal group is the group of global conformal transformations $\mathbb{R}^{2,0}\to \mathbb{R}^{2,0}$.
2. The Lie algebra of the conformal group is finite-dimensional.
3. For each $x\in \mathbb{R}^{2,0}$, we have an algebra of "infinitesimal conformal transformations" (which I believe can be interpreted as conformal Killing fields) defined in a neighborhood of $x$, which is the Witt algebra. This is infinite-dimensional.
4. The conformal Killing fields generate the local conformal transformations $U\to U'$.

So it seems that locally we get Virasoro algebras of Killing fields, and these generate local conformal transformations, even though there is no Lie group whose Lie algebra is the Virasoro algebra.

How far off am I?