actually, yes, it does mean that:so if the vector field is zero at p, it does not mean the flow is contant at p?
let X be the vector field in question such that X(p)=0.
By definition the flow is the unique path of local diffeomorphims f_t given pointwise in the domain of X by the differential equation (d/dt)(f_t (q))=X(q) when t=0 with initial condition that f_0(q)=q at all points q in the domain of X.
Clearly, if X(p)=0, then f_t(p)=f_0(p)=p.
This is the reason why I kept pointing out that the flow is a path of diffeomorphisms, not just one diffeo. For explain's attempted counterexample, the flow is simply the path of rotations about the origin on the plane, all of which of course fix the origin.