...The main difference between the two is, Black Star is supposed to only be able to capture black souls, and can't handle white souls, vice versa with Azura's Star...
Wrong. Azura's Star and the Black Star are exactly the same EXCEPT that the Black Star can ALSO capture "black" souls (player race souls) in addition to "white" souls. That is the only difference between the two. Both AS and BS follow the same rules as all other soul gems. The game will try to fit the soul into the appropriate gem. If there is no empty soul gem that matches the soul, it will put the soul in the next higher empty soul gem. So if AS or BS are the only empty soul gems you have, and you soul trap a skeever, it will place that soul in AS/BS. If you have the Black Star and attempt to soul trap a Grand "White" soul, it will fill the Black Star just as it would Azura's Star. The game will try to avoid putting a white soul in a black soul gem (and the Balck Star), but if it is the only empty soul gem available, it will fill it with whatever soul you trapped.