The alpha star in a constellation has the highest brightness, but does that mean it has the highest flux density among the other stars in the constellation?
"Brightness" can mean two different things. The apparent brightness of the point of light in the sky as viewed from Earth and the absolute brightness of the star as a physical object, viewed from a standard distance.
For example Eta Carina is a very very very bright star in absolute terms, but only an averagely bright point of light. Sirus is the brightest star in the sky, but far less powerful. Sirius appears brighter than Eta Carina because it is closer.
The Bayer designations (Alpha, Beta etc) are based, in part, on the apparent brightness. The flux density is a measure of the apparent brightness. So Sirius has the highest flux density.