We investigate magnetic reconnection dynamics using three simulation codes to isolate ion kinetic effects from Hall effects: a full hybrid code, a Hall MHD code, and a new Hall-less hybrid code. The structure of the quadrupole magnetic field (also known as the core field or out-of-plane field) is used as the physical quantity for this comparison because it is a good proxy of fast magnetic reconnection. Examination of the core field time evolution equations for the Hall-less hybrid and Hall MHD models reveal inherent differences between these models. In particular, it demonstrates that core fields can be generated in the kinetic regime even in the absence of the Hall term. We attribute this result to ion kinetic effects, e.g., ion diamagnetic drifts driven by an anisotropic ion stress tensor. This result is borne out by our simulations. Additionally, kinetic ion dynamics leads to core fields with widths that can extend to several ion inertial lengths in the vicinity of X-line and tens of ion inertial lengths further away. In contrast, in Hall MHD, the typical case is about an ion inertial length in the vicinity of the X-line and a few ion inertial lengths further away. These results pose challenges for improving fluid models of reconnection.