My pocket book shows the BK2408 with foot, axial and radial mounting features.
It says BK24 are 3-phase .
BS are classed as split-phase, BC capacitor start.
Some are shown with hand reset, or auto-reset units mounted on the terminal cover.
No mention of S anywhere.
The standard connection shows Main winding across A1-A2.
Aux (start winding ) in series with centrifugal switch also across A1-A2.
Whether you need a capacitor ,or not, for reversing will depend on it being a split phase motor ?