In an extended superconductor-topological insulator-superconductor (S-TI-S) Josephson junction in a magnetic field, localized Majorana bound states (MBS) are predicted to exist at the cores of Josephson vortices where the local phase difference across the junction is an odd-multiple of $\pi$. These states contribute a supercurrent with a $4\pi$-periodic current-phase relation (CPR) that adds to the conventional $2\pi$-periodic sinusoidal CPR. In this work, we present a comprehensive experimental study of the critical current vs. applied magnetic field diffraction patterns of lateral Nb-Bi$_2$Se$_3$-Nb Josephson junctions. We compare our observations to a model of the Josephson dynamics in the S-TI-S junction system to explore what feature of MBS are, or are not, exhibited in these junctions. Consistent with the model, we find several distinct deviations from a Fraunhofer diffraction pattern that is expected for a uniform sin$({\phi})$ CPR. In particular, we observe abrupt changes in the diffraction pattern at applied magnetic fields in which the current-carrying localized MBS are expected to enter the junction, and a lifting of the odd-numbered nodes consistent with a $4\pi$-periodic sin$(\phi/2)$-component in the CPR. We also see that although the even-numbered nodes often remain fully-formed, we sometimes see deviations that are consistent with quasiparticle-induced fluctuations in the parity of the MBS pairs that encodes quantum information.
