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We present Josephson interferometry using high-quality Corbino-geometry Josephson junctions fabricated on a surface of a bulk-insulating three-dimensional (3D) topological insulator (TI). The Josephson junction formed with a 3D TI surface as the weak link can be used for probing signatures of Majorana bound states (MBSs) by measuring the Josephson current-phase relation (CPR). We fabricate edge-free Corbino-style Josephson junctions on Sn-doped Bi_1.1Sb_0.9Te₂S (Sn-BSTS) single crystals with an extremely low bulk carrier concentration. We preserve atomically flat pristine surfaces of the TI by Te thin film deposition using ultrahigh-vacuum molecular beam epitaxy. The excellent quality of the Josephson coupling in our devices is evidenced by a high transition temperature, a large I_cR_N product and induced gap, and the observation of multiple Andreev reflections and well-defined Fraunhofer patterns. These observations are attributed to the highly transparent interface between the high-quality Sn-BSTS channel and an induced superconductor under Nb contacts. Moreover, we employ different geometries of Corbino-style Josephson junctions to enable Josephson interferometry within a single junction formed on a TI surface and show the important role of geometry in capturing characteristics of the CPR that contain signatures of MBSs.