Sorry, I will edit.

I see the problem now, thank you very much!

I am sorry, but there a still a couple of things I still don't get. I'll write here, there may be naive mistakes, please correct me if I am wrong. $PMCG(S_g^b)$ is generated by Dehn twists around non-trivial simple closed curves, right? Every non-trivial simple closed of $S_g^b$ can be seen as a non-trivial simple closed curve in $S_{g,b}$, so it seems that $PMCG(S_g^b)$ is "naturally" a subgroup of $PMCG(S_{g,b})$. So why does the inclusion $PMCG(S_{g}^b) \to PMCG(S_{g,b})$ not split the sequence above? Thank you in advance.

Thank you for your answer. Could you clarify your point about abelianizations and precise why the abelianization of $PMCG(S_{g,b})$ is trivial?

You're right. No multiple edges between two points in the interior of the disk. I'll edit.

I know that paper, it's about triangulations of a polygon. Labelled points are inside the disk in my case. Methods used are pure hyperbolic geometry (!)