How should the magnetic force between two magnets be calculated effectively?

I have read this Wikipedia page, but I can’t understand it.

In this picture from that page, in the formula, there is just one flux density, but what about the flux density of the second magnet? What if the magnets aren’t identical?

Also, why do we have both area and radius in the formula? Aren’t the two terms proportional? And also, why there are both $mu_0$ and $B_0$? There is $mu_0$ in the $B_0$ formula too!

Another thing that I can’t get, is that what’s the difference between a cylindrical bar magnet and a cylindrical magnet? How are these separated into two different classes with different formulas? Even the latter needs elliptical integrals and magnetization which are very hard.