Since I erred in my initial comment, to compensate here is a characteristic-free argument over general fields $k$ (that also adapts to work over general commutative rings). As in Paul Levy's answer, we will show that beyond the trivial rank-1 case, PGL$_2$ is obtained precisely for the short root of adjoint type B$_n$ with $n \ge 2$ (and one could make the statement more uniform including rank 1 via type B$_n$ for $n \ge 1$ if we declare that the roots for type ${\rm{B}}_1 = {\rm{A}}_1$ are short, or perhaps both long and short).

The simply connected case always gives ${\rm{SL}}_2$ for the general reason that the derived group of any torus centralizer in a simply connected semisimple group is always simply connected.  A proof of this general fact, based on the characterization of "simply connected" in terms of simple positive coroots being a basis of the cocharacter lattice, is sketched in Corollary 9.5.11 of https://www.ams.org/open-math-notes/omn-view-listing?listingId=110663
(where the argument is given over fields, and a reference is provided to make the argument work over rings).

Now consider a general split connected semisimple $k$-group $G$ that is (absolutely) simple with split maximal $k$-torus $S$, and let $f:\widetilde{G} \rightarrow S$ be the split simply connected central cover. The preimage $\widetilde{S} := f^{-1}(S)$ is a split maximal $k$-torus of $\widetilde{G}$, and $\Phi(G,S)= \Phi(\widetilde{G},\widetilde{S})$ via the finite-index inclusion ${\rm{X}}(S) \subset {\rm{X}}(\widetilde{S})$
due to the centrality of the subgroup scheme $\ker f$ (even though ${\rm{Lie}}(f)$ may not be surjective).  

Here is the main trick to avoid too much case-work: by the settled simply connected case we know $[\widetilde{G}_a, \widetilde{G}_a] = {\rm{SL}}_2$ for all $a \in \Phi$, and this maps onto $[G_a,G_a]$ via a central isogeny. In particular, if the group scheme $\ker f$ has *odd* degree then we're done (as SL$_2$ has scheme-theoretic center $\mu_2$ of order 2).  Thus, the case of type A$_2$ (i.e., PGL$_3$ and SL$_3$) always gives SL$_2$ since 3 is odd. This will dispose of nearly all remaining cases below because most vertices in Dynkin diagrams are adjacent to one with the same length.

Let $n>1$ be the rank of the reduced and irreducible root system $\Phi$. We may assume that our root $a$ belongs to a chosen basis $\Delta$ of $\Phi$. We may also assume $G$ is not simply connected or else there's nothing to do, so we're not in types G$_2$ and F$_4$.  We shall now show that we get SL$_2$ away from the short roots of B$_n$ and the long roots of type C$_n$ (and then we will analyze these remaining cases). Being away from those two classes of cases, note that we're not in type ${\rm{B}}_2 = {\rm{C}}_2$. Thus, $n \ge 3$ and in the Dynkin diagram the root $a \in \Delta$ is adjacent to another root $b \in \Delta$ with the *same* length.  Thus, the codimension-2 subtorus $S_{a,b} = (\ker a \cap \ker b)^0_{\rm{red}} \subset S$ killed by $a$ and $b$ has centralizer whose derived group $G_{a,b}$ is of type A$_2$ (with split maximal $k$-torus $S' = S \cap G_{a,b} = a^{\vee}(\mathbf{G}_m)b^{\vee}(\mathbf{G}_m)$ and Dynkin diagram having nodes $a|_{S'}, b|_{S'}$). The original commutator subgroup of interest for $(G, S, a)$ agrees with the one for $(G_{a,b}, S', a|_{S'})$. This reduces our task to the settled case of type A$_2$!

It remains to consider long roots of type C$_n$ with $n \ge 2$ and short roots of type B$_n$ with $n \ge 2$. We will reduce these to consideration of adjoint type ${\rm{B}}_2 = {\rm{C}}_2$ (concretely, SO$_5$). Using the action of the Weyl group, we may arrange that our root $a$ corresponds to the unique node that is long for type C and short for type B. Let $b \in \Delta$ be the unique node adjacent to $a$ in the diagram. The center of simply connected type B$_n$ is $a^{\vee}(\mu_2)$ (this is the key fact, as also noted in Paul Levy's comment and answer), so by open cell considerations we see that the group $G_{a,b}$ made as above but now for the nodes $a$ and $b$ as just defined is also of adjoint type when we are in type B. 

Keeping in mind what we are aiming to show (that we get SL$_2$ for the long roots of adjoint type C$_n$ with $n \ge 2$ and PGL$_2$ for the short roots of adjoint type B$_n$ with $n \ge 2$), since the simply connected case gives only SL$_2$ in general and in types B and C the only options are simply connected and adjoint type (as the fundamental group of the root system has order 2) we can pass to $G_{a,b}$ to reduce to the case of adjoint type ${\rm{B}}_2={\rm{C}}_2$. (It doesn't matter that for type C this passage to rank 2 might leak back to the simply connected case, since that case always gives SL$_2$ anyway.)

Letting $\{a, b\}$ be a basis for type ${\rm{B}}_2 = {\rm{C}}_2$ with $a$ short and $b$ long, we want to show that $[G_a, G_a] = {\rm{PGL}}_2$ and $[G_b, G_b] = {\rm{SL}}_2$. The center of $\widetilde{G}$ is $a^{\vee}(\mu_2)$ as noted already (contained in the split maximal torus $a^{\vee}(\mathbf{G}_m)$ of $[\widetilde{G}_a, \widetilde{G}_a] = {\rm{SL}}_2$), so for $G = \widetilde{G}/Z_{\widetilde{G}} = \widetilde{G}/a^{\vee}(\mu_2)$ we have $[G_a,G_a] = [\widetilde{G}_a,\widetilde{G}_a]/a^{\vee}(\mu_2) = {\rm{SL}}_2/\mu_2 = {\rm{PGL}}_2$.  Likewise, $[\widetilde{G}_b,\widetilde{G}_b]$ meets the rank-2 split maximal torus $\widetilde{S} = \mathbf{G}_m^{\Delta^{\vee}}$ in precisely its own split maximal torus $b^{\vee}(\mathbf{G}_m)$, so its intersection with $a^{\vee}(\mathbf{G}_m)$ is trivial. In particular, its intersection with $a^{\vee}(\mu_2)$ is trivial, so the central isogeny ${\rm{SL}}_2 = [\widetilde{G}_b, \widetilde{G}_b] \twoheadrightarrow [G_b,G_b]$ has trivial (scheme-theoretic) kernel and thus is an isomorphism.