"A Primer on Mapping Class Groups" wrote

Let $\mathrm{Homeo}_+(S, \partial S)$ denote the group of orientation-preserving homeomorphisms of $S$ that restrict to the identity on $\partial S$. $\mathrm{Mod}(S)$ is the group of isotopy classes of elements of $\mathrm{Homeo}_+(S, \partial S)$, where isotopies are required to fix the boundary pointwise.

and

Let $\mathrm{PMod}(S_{g,n})$ denote the pure mapping class group of $S_{g,n}$, which is defined to be the subgroup of $\mathrm{Mod}(S_{g,n})$ consisting of elements that fix each puncture individually.

What is the difference between the two definitions? Doesn't "restrict to the identity on $\partial S$" mean for for all $x\in \partial S$ we have $\phi(x)=\mathrm{id}(x)=x$, which implies $\phi$ fixes each puncture individually?