As Kevin says, the "right" definition of ${\rm{PSL}}_n$ is as representing the quotient sheaf ${\rm{SL}}_n/\mu_n$, just as one defines ${\rm{PSO}}(q) = {\rm{SO}}(q)/Z_{{\rm{SO}}(q)}$ (with $Z_G$ denoting the scheme-theoretic center of a smooth group $G$). So really, there is no difference between ${\rm{PSL}}_n$ and ${\rm{PGL}}_n$ when defined correctly, and likewise ${\rm{PSO}}(q) = {\rm{PGO}}(q)$. Personally, I avoid the notation ${\rm{PSL}}_n$ like the plague, since it creates too much confusion.

Lest this seem like a flippant answer, let me point out that for a general ring $R$ with nontrivial Picard group, it is likewise not true that ${\rm{PGL}}_n(R)$ is the "naive" thing either! For example, if $R$ is a Dedekind domain whose Picard group has nontrivial 2-torsion then ${\rm{PGL}} _2(R)$ is generally bigger than
${\rm{GL}} _2(R)/R^{\times}$. And this is not a quirk with algebraic geometry. The same thing happens with Lie groups: if a smooth manifold $M$ has nontrivial 2-torsion line bundles it can and does happen that there are $C^{\infty}$ maps $f:M \rightarrow {\rm{PGL}} _2(\mathbf{R})$ which do not arise from a map to ${\rm{GL}} _2(\mathbf{R})$ (concretely, pulling back the quotient map ${\rm{GL}} _2(\mathbf{R}) \rightarrow {\rm{PGL}} _2(
\mathbf{R})$ along $f$ yields a line bundle on $M$ that may be non-trivial).

And the "weirdness" of it all (based on experience over an algebraically closed field) is also seen by the fact that the concrete definition of the group scheme ${\rm{PGL}}_n$ is as a basic affine open in the projective space of $n \times n$ matrices, and we know that "points" of projective spaces are a subtle thing (compared with the case of geometric points) when the source has nontrivial line bundles (whether a scheme or manifold).

Since one cannot get by with field-valued points when doing representability arguments, the same "problem" which one sees for the naive viewpoint on ${\rm{PSL}} _n$ is also relevant when doing proofs for ${\rm{PGL}} _n$. The difference is that for the latter one has to work more "globally" to see the surprise because the degree-1 Zariski and fppf cohomologies for $\mathbf{G} _m$ coincide (so for local rings nothing funny happens, as they have vanishing higher Zariski sheaf cohomology) whereas for the former there is already a funny thing happening for local rings and even fields (which can have nontrivial degree-1 fppf cohomology for $\mu_n$). In more concrete terms, it is equivalent to define the functor ${\rm{PGL}} _n$ as a quotient sheaf for either the Zariski or fppf topologies, whereas for ${\rm{PSL}} _n$ one has to sheafify for the fppf topology (etale ok when $n$ is a unit on the base), and in either case the naive functor on rings (inspired by the case of algebraically closed fields) is not even a Zariski sheaf and hence beyond local rings something has to be done to get the right functor (e.g., one that is representable).

Many books on linear algebraic groups use a version of algebraic geometry that is not well-suited to the subtleties of quotient considerations (e.g., Borel uses Serre's clever method "quotient by $p$-Lie algebra" to handle quotients by infinitesimal groups without saying "infinitesimal group", and some of his quotient arguments would be much shorter if he could have used flatness systematically). In particular, Springer's book has some serious errors when the ground field is not algebraically closed (in the later parts, where he discusses $F$-reductive groups and related things). For example, he uses the incorrect argument that surjectivity of an $F$-map between smooth $F$-varieties can be checked on $F_s$-points, which is not true when $F$ is not perfect (remove an inseparable point from the affine line and consider the inclusion into the line). So be careful in that part of his book. (Some statements are false, not just proofs.)