I've seen two definitions of the "order" of an integrator for ordinary differential equations. The first (more common) one is that the method has order $k$ if the error introduced after one timestep $h$ is $O(h^{k+1})$. The second is that the method has order $k$ if it integrates a polynomial of degree $\le k$ exactly. For most common integrators (e.g., Runge-Kutta) the two definitions are equivalent.
Is there a general theorem on what conditions must be satisfied for the two definitions to be equivalent?