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The growth of α-Ga2O3 and α-(InxGa1−x)2O3 on m-plane α-Al2O3(101̄0) by molecular beam epitaxy (MBE) and metal-oxide-catalyzed epitaxy (MOCATAXY) is investigated. By systematically exploring the parameter space accessed by MBE and MOCATAXY, phase-pure α-Ga2O3(101̄0) and α-(InxGa1−x)2O3(101̄0) thin films are realized. The presence of In on the α-Ga2O3 growth surface remarkably expands its growth window far into the metal-rich flux regime and to higher growth temperatures. With increasing O-to-Ga flux ratio (RO), In incorporates into α-(InxGa1−x)2O3 up to x ≤ 0.08. Upon a critical thickness, β-(InxGa1−x)2O3 nucleates and, subsequently, heteroepitaxially grows on top of α-(InxGa1−x)2O3 facets. Metal-rich MOCATAXY growth conditions, where α-Ga2O3 would not conventionally stabilize, lead to single-crystalline α-Ga2O3 with negligible In incorporation and improved surface morphology. Higher TTC further results in single-crystalline α-Ga2O3 with well-defined terraces and step edges at their surfaces. For RO ≤ 0.53, In acts as a surfactant on the α-Ga2O3 growth surface by favoring step edges, while for RO ≥ 0.8, In incorporates and leads to a-plane α-(InxGa1−x)2O3 faceting and the subsequent (2̄01) β-(InxGa1−x)2O3 growth on top. Thin film analysis by scanning transmission electron microscopy reveals highly crystalline α-Ga2O3 layers and interfaces. We provide a phase diagram to guide the MBE and MOCATAXY growth of single-crystalline α-Ga2O3 on α-Al2O3(101̄0).