Why is Rigel called the 'beta star' in Orion and not Betelgeuse which is fainter?

The problem with Rigel and Betelgeuse is that they are practically the same brightness; Rigel has a visual magnitude of +0.14m and Betelgeuse is at +0.41.

When Johannes Bayer invented the 'alpha, beta, gamma...' system for ranking stars visually, he did the best he could, but clearly chose wrong and the designation he gave Rigel as the 'beta star' is still used. This is a well-known 'mistake' made by him back then...but an honest mistake, since he had no instruments other than the naked eye, that could make actual brightness measurements of stars!