A novel strategy for extracting axial (AV) and toroidal (TV) vortices in the magnetically-induced current density (MICD) in molecular systems is introduced, and its pilot application to LiH molecule is demonstrated. It exploits differences in the topologies of AV and TV cores and involves two key steps: selecting a scalar function that can describe vortex cores in MICD and its subsequent topological analysis. The scalar function of choice is $\Omega^{B_\alpha}$ based on the velocity-gradient $\Omega$ method known in research on classical flows. The Topological Data Analysis (TDA) is then used to analyze the $\Omega^{B_\alpha}$ scalar field. In particular, TDA robustly assigns distinct topological features of this field to different vortex types in LiH: AV to saddle-maximum separatrices which connect maxima to 2-saddles located on the domain's boundary, TV to a 1-cycle of the super-level sets of the input data. Both are extracted as the most \emph{persistent} features of the topologically-simplified $\Omega^{B_\alpha}$ scalar field.