As a general rule you should be able to tell bead cultured pearls from natural or non-bead cultured pearls by what you see down the drill hole. Since the vast majority of bead nuclei are white FW shell what you should see is a boundary between the shell nucleus and pearl nacre overgrowth with a faint yellow to white graduation (white pearls) from the surface the deeper/thicker the nacre is until you reach the boundary and then the centre should look more white or cream (even colouration throughout). If you try and look down the drill hole from the other side you will see the same, perhaps more clearly, perhaps less clearly. Just have try from both end of the drill hole. Natural pearls on the other hand are a graduated white/cream/light yellow/yellow/orange or black when you look down the drill holes. You cannot really use this technique to separate natural from non-bead cultured as they are similar. It only helps with BC v. natural/NBC. Try on some loose pearls and see if you can put the theory into practice. Hope it helps.
The cleaning really did help improve your necklace a lot. The broken pearl at one end should allow you to see the internal structure of that one pretty clearly!