http://math.stanford.edu/~conrad/papers/luminysga3.pdf, appendix C.2 defines $ \mathrm{SO}(q) $ as the kernel of the algebraic group morphism $ D_q: \mathrm{O}(q) \to (\mathbb Z / 2 \mathbb Z)_S $, which is defined using Clifford algebras. Conrad claims that the usual, naive definition, in terms of the determinant, yields a group scheme that is not flat and/or smooth in characteristic $ 2 $ (or over schemes $ S $ where $ 2 $ is not a unit), while the kernel of $ D_q $ is smooth over $ S $.
However, I tried to compute explicitly what $ D_q $ is, and the equations describing it seem to boil down to the exact same restriction on determinants. So how is this a better definition and how does it yield better properties for $ \mathrm{SO}(q) $?