Let $G$ be a group. In An introduction to homological algebra, Chapter 6.9 Weibel defines a universal central extension as a central extension $$0 \to A \to X \to G \to 1,$$ which is initial with respect to all central extensions. In other words, if $0 \to B \to Y \to G \to 1$ is any other central extension, then there is a unique homomorphism $X \to Y$ which is compatible with the two maps to $G$.
The first basic lemma (6.9.2) Weibel proves is that if there exists a universal central extension, then both $X$ and $G$ are perfect (i.e. $[G,G] = G$).
Now both Wikipedia and this notes (Exercise 10) claim that the extension $$0 \to \mathbb Z \to \mathcal B_3 \to \operatorname{PSl}_2(\mathbb Z) \to 1$$ is universal. (For the definition of the extension, see my other question.)
But that apparently contradicts Weibel's lemma, because the group $\mathcal B_3$ is not perfect, it's abelization is $\mathbb Z$ (by exercise 7 of the above notes). So I guess there is a different notion of universal central extension? This section in Wikipedia seems to also indicate that, but I don't quite understand what they mean, and it seems to mostly care about finite groups? Does that make a difference?
This question had been discussed at this MO-post. The "usual" definition of a universal central extension $X$ of a group $G$ assumes that $G$ is perfect, and that it is the unique (up to isomorphism) group $X$ that is a Schur covering group of $G$. In other words, the universal central extension of a perfect group is also perfect - see here.
However, the terminology has been also used as follows: In a universal central extension $1 \to M \to X \to G \to 1$ of $G$, the image of $M$ in $X$ is required to be in the commutator subgroup $[X,X]$ of $X$. In our case, the intersection of the central cyclic subgroup $\langle x^2 \rangle = \langle y^3 \rangle$ of $B_3$ with $[B_3,B_3]$ is trivial.