In the complex analytic setting, it is easy to see that the moduli space of a sphere with four punctures is $\mathcal{M}=\mathbb{CP}^1 / { 0,1,\infty }$, since I can use a Moebius transformation to send three of the punctures at those location whilst the fourth puncture is free to move.

On the other hand, the moduli space can be obtained as the quotient of the Teichmüller space by the mapping class group. Teichmüller space can be described using e.g. Fenchel-Nielsen coordinates, for each simple closed curve on the sphere we have two coordinates (a length and an angle). The mapping class group can be described using Dehn twists.

I always assumed that the three boundary points ${0,1,\infty}$ could be interpreted as sending to zero the length of three distinct curves separating the punctures into one of the three sets $(12)(34)$, $(23)(41)$ or $(13)(24)$, but I have recently realized that these curves are related to each other by suitable elements of the mapping class group. So I would naively say that the corresponding limits should be identified as well.

How do I reconcile the above picture of the moduli space with the one coming from Teichmüller modulo mapping class group? How do I see three boundary points in this picture?