Notions of Fundamental Groups for semisimple algebraic groups

Let $$G$$ be a connected, semisimple linear algebraic group over a field $$k$$. In Springer's Linear Algebraic group, the Fundamental Group of $$G$$ is defined by a certain quotient of groups $$P/Q$$, which is defined via the root system $$(X,R)$$ of $$G(k^{alg})$$ (Springer, 8.1.11). Here $$k^{alg}|k$$ is an algebraic closure, $$P:=\{v\in X\otimes \mathbb{R}~|~\langle v,R^\vee\rangle\subset\mathbb{Z}\}$$ is the weight lattice and $$Q\subset X$$ is the subgroup generated by $$R$$, the root lattice. In Milne's Algebraic Groups the Fundamental Group of $$G$$ is defined as the kernel of the universal/simply connected covering $$G_{sc}\rightarrow G$$ (Milne(online version), 20.5). Now in (Springer, 17.3.1), it is said that the root system corresponding to the Dynkin-classes $$D_n$$ have Fundamental Group with order $$4$$ ($$n\ge 4$$)). But also in (Springer, 17.3.1) for $$char(k)\neq 2$$, a $$k$$-split simple group in class $$D_n$$ is given by (some) $$SO_{2n}$$. In Serre's Galois Cohomology (III §3.2 b)) the universal cover for $$SO_{2n}$$ ($$n\ge 2$$) is given by $$0\rightarrow\mu_2\rightarrow Spin_{2n}\rightarrow SO_{2n}\rightarrow 0,$$ so the (group of $$k$$-valued or $$k^{alg}$$-valued points of the) Fundamental Group by Milne has order $$2$$.

What is the connection between these notions? Or am I missunderstanding something?

If $$(G,T)$$ is a split reductive group, then $$X^*(Z))=X^*(T)/\mathbb{Z}\Phi$$, where $$Z$$ is the center of $$G$$ and $$\mathbb{Z}\Phi$$ the submodule generated by the roots (Milne 2017, 21.6). When $$G$$ is a simply connected semisimple group, this becomes $$X^*(Z)=P/Q$$ (ibid. 24c); in this case, $$G$$ is the universal covering group of the adjoint group $$G/Z$$ and $$Z$$ is the fundamental group of $$G/Z$$.