The analytical jacobian is directly differential from forward kinematic, and geometric jacobian is considered the geometric relation. The important point is that angular velocity does not have an integral like associated term like "angular position".

differential quantities in the operational space.

quantities of clear physical meaning.

Note that $$\dot x = T(x) v$$ where T is a transformation matrix that is a function of $$x$$ only. $$T = \begin{pmatrix} I && 0 \\ 0 && R \end{pmatrix}$$ where $$R$$ is the end-effector orientation. Then we can write $$J_A(q) = T(x)J(q)$$.