The packings and defects of the strongly coupled two-dimensional Coulomb clusters with particle number N from a few to a few hundred with different forms of mutual repulsion and central confining potentials at zero temperature are investigated using molecular-dynamics simulation through many annealing cycles. The circular symmetry of the confining potential and the interplay with the mutual repulsion lead to the strong competition between the outer circular shells and the inner triangular lattice. Generic packing behaviors, such as the concentric shells with the classical periodic packing sequence at small N, and the triangular latticelike inner core surrounded by a few outer circular shells at large N are observed. The effects of changing the interaction and confining potentials on the detailed packing sequence, the radial variation of packing density, and the positions of the shell-triangular core interface are investigated with a detailed study of the cluster structures along with the formation and distribution of topological defects.