Magnetic skyrmions are nanoscale magnetic whirls that can be driven by currents via spin torques. They are promising candidates for spintronic devices such as the racetrack memory, where a motion along the uniform current is typically desired. However, for spin torque nano-oscillators in Corbino disks, the goal is to achieve a circular motion, perpendicular to the radially applied current. As we show, based on analytical calculations and micromagnetic simulations, Bloch skyrmions engage in a circular motion with frequencies in the MHz range when driven by spin-orbit torques. In contrast, Néel skyrmions get stuck at the edges of the disk. Our analysis reveals that the antagonistic dynamics between Bloch- and Néel-type magnetic textures arise from their helicity. Furthermore, we find that skyrmioniums, which are topologically trivial variations of skyrmions, move even faster and allow an increase in the current density without being pushed toward the edges of the disk. When driven by spin-transfer torques instead, Bloch and Néel skyrmions no longer exhibit different dynamics. Instead, they move along a circular trajectory due to the skyrmion Hall effect caused by their topological charge. Consequently, the topologically trivial skyrmioniums inevitably become trapped at the disk edge in this scenario. To provide a comprehensive understanding, our paper also examines currents applied tangentially, further enriching our insights into skyrmion dynamics and appropriate current injection methods for skyrmion-based devices.