We show experimentally through single-shot time-resolved conductance measurements that magnetization reversal through domain wall motion in sub-100 nm diameter magnetic tunnel junctions is dominated by two distinct stochastic effects. The first involves the incubation delay related to domain wall nucleation, while the second results from stochastic motion in the Walker regime. Micromagnetics simulations reveal several contributions to temporal pinning of the wall near the disc center, including vertical Bloch line nucleation and wall precession. We show that a reproducible ballistic motion is recovered when Bloch and Néel wall profiles become degenerate in energy in optimally sized discs, which enables quasi-deterministic motion.