Looking for an answer to https://physics.stackexchange.com/q/757691/, here
I came across the notion of universal central extensions, let me call this definition 1:
The group $G^{\star}=\mathbb{R}^{N} \times_{\xi^{ \star}} \widetilde{G}$ defined as a central extension of the universal cover $\widetilde{G}$ by $\mathbb{R}^{N}$, i.e. as the product manifold $\mathbb{R}^{N} \times \widetilde{G}$ endowed with the operation \begin{equation} (a,x)(b,x)=(a+b+\xi^{\star}(x,y),xy), \end{equation} for $a,b \in \mathbb{R}^{N}$ and $x,y \in \widetilde{G}$, is called the universal central extenson of $G$. Given the covering map $\widetilde{p}:\widetilde{G} \to G$, we define the universal covering map in an obvious way \begin{equation} p^{\star}:G^{\star} \to G;\quad (a,x) \mapsto \widetilde{p}(x). \end{equation}
However, in mathematical texts, a universal central extension is defined as a central extension, through which every other central extension factors uniquely. Let me refer to this as definition 2.
My question is: are definition 1 and definition 2 equivalent, namely: is it true that $1 \implies 2$ and $2 \implies 1$?
The problem here is that "universal" in the context of central extensions can mean different things when used by different authors, and that in contrast to other "universal" constructions in mathematics, the universal central extension for $\mathrm{U}(1)$-extensions may not be unique. A paper that discusses this in detail is Raghunathan's "Universal Central Extensions", itself the appendix to the equally worthwhile "Symmetries and Quantization: Structure of the State Space" by Divakaran.
As for the construction in the question, it is meant to be universal in the following sense: For every $\mathrm{U}(1)$-extension $e_\mu : G_\mu \to G$ with $\mu\in H^2(G,\mathrm{U}(1))$ classifying the extension, there is a preimage $\chi_\mu\in \hat{K}$ for $K = \ker(p^\ast) \cong \mathbb{R}^N \times\pi_1(G)$ (theorem 6.8 in the paper by Nesta) and so we get a map $$q_\mu : G^\ast = \mathbb{R}^N\times_{\xi^\ast} \tilde{G} \to \mathrm{U}(1)\times_\mu G \cong G_\mu, (x,g)\mapsto (\chi_\mu(x), \pi(g))$$ with $p^\ast = q_\mu \circ e_\mu$, i.e. $p^\ast : G^\ast \to G$ factors through every $\mathrm{U}(1)$-extension $e_\mu$. The map $q_\mu$ is unique exactly when the morphism $\hat{K}\to H^2(G,\mathrm{U}(1))$ from theorem 6.8 is an isomorphism.