We study hydrogen-bond dynamics in liquid water at low temperatures using molecular dynamics simulations. We analyze the dynamics using energetic and geometric definitions of a hydrogen bond, and employ two analysis methods: (i) a history-dependent correlation function, related to the distribution of bond lifetimes, and (ii) a history-independent correlation function. For method (i) we find an approximately Arrhenius temperature dependence of the bond lifetime, and find that the distribution of bond lifetimes is extremely sensitive to the choice of bond definition. For method (ii) we find-independent of bond definition-that the dynamics are consistent with the predictions of the mode-coupling theory, suggesting that the slow dynamics of hydrogen bonds can be explained in the same framework as standard transport quantities. Our results allow us to clarify the significance of the choice of both bond definition and analysis technique.