Different definitions for the Teichmüller space of puctured spheres

Suppose that $S$ is a compact connected oriented surface. Suppose that $P \subset S$ is a collection of marked points (none on the boundary). Let $g = g(S)$ be the genus of $S$. Let $p = |P|$ and let $b = |\partial S|$.

The space given in the Primer has real dimension $6g - 6 + 2p + 3b$. The space given in Douady-Hubbard has real dimension $-6 + 2p$.

There is no way to "turn boundary components into punctures" or conversely. This is because the the conformal structure of the interior of a surface with geodesic boundary "remembers" the length of the geodesic boundary.

If you disallow boundary and genus, then you are comparing spheres with marked points to spheres with punctures (with conformal structures and hyperbolic metrics, respectively).

(1) From a marked conformal sphere, the uniformisation theorem produces a unique hyperbolic metric on the punctured sphere.

(2) The Riemann removable singularities theorem produces the desired conformal structure by "continuing" the complex structure to the missing point.

(3) The mapping class groups are the same if you make the same decision in both cases about how homeomorphisms to permute marked points (respectively, permute the ends of the surface coming from punctures).