We investigate the effect of a magnetic field on the shape and director field of nematic droplets in dispersions of sterically stabilized and charge-stabilized colloidal gibbsite platelets with a negative diamagnetic anisotropy. Depending on the magnetic field strength and tactoid size, we observe with polarized light microscopy several interesting structures, with different shapes and director fields both with and without defects. In particular, our findings provide the first experimental evidence for the existence of the split-core defect structure predicted ten years ago by Mkaddem and Gartland [Phys. Rev. E 62, 6694 (2000)]. The split-core structure is a metastable director-field configuration that can be stabilized by a sufficiently strong externally applied magnetic field but only if the diamagnetic anisotropy of the particles is negative. To account for our observations, we present a calculation of the stability regions of different shapes and director-field structures as a function of tactoid size, anchoring conditions, surface tension, elastic constants, and magnetic field strength. By fitting the experimental data to the theoretically predicted structures, we are able to extract values for the splay elastic constant, interfacial tension, and anchoring strength. Remarkably, we find significant differences between the two systems studied: for sterically stabilized gibbsite in bromotoluene the anchoring strength is one order of magnitude larger than that of aqueous gibbsite, with the latter exhibiting weak and the former strong anchoring of the director field to the interface. The splay elastic constants that we obtain are in agreement with earlier experiments, simulations, and theory, while the interfacial tension and anchoring strength are considerably larger than what was found in earlier experiments.