We apply the Halperin-Mazenco formalism within the Gross-Pitaevskii theory to characterize the kinematics and nucleation of quantum vortices in a two-dimensional stirred Bose-Einstein condensate. We introduce a smooth defect density field to measure the superfluid vorticity, a topologically conserved quantity. We use this defect density field and its associated current density to study the precursor patterning that occurs within the repulsive potential of obstacles to determine the onset of eddy nucleation and shedding. We show that phase slips form within hard potentials even without vortex nucleation, whereas for soft potentials phase slips occur only above the critical stirring speed leading to vortex nucleation. The Halperin-Mazenco formalism provides an elegant and precise way to directly derive the point vortex dynamics from the Gross-Pitaevskii equations.