My reading suggests there is a wide variation of grit size and uniformity between sources with the same name such as "white diamond". Since you are polishing to achieve a uniform fine spacing of very fine scratches, you are likely to get less consistency from the general hardware or "Borg" products. This is one of those -- pay your money and take your chances situations.
I have been down that path and did find that a few oversize pieces of grit in the product made the effort largely unsuccessful and went to specialty sources with better uniformity. Since a sitck of white diamond from a specialty house will last about three lifetimes, the cost per use is pretty small.
Note that there are also specialty polishes with high control of uniformity with the use of plastics. There is good cross walk with the specialty products on auto finish polishing too. Auto finish is all about very very fine uniform surfaces.
White diamond generically is used as a metal polish for such things as stainless, and with metals, the grit consistency is not as important as it is for a much softer material such as wood and plastics.