Here is the best explanation of the difference between Biewer Terriers and Yorkshire Terriers that I can find, because at least it is documented by MARS testing documents, even if the figures are low resolution:
Evaluation of DNA Changes in a New Dog Breed
The conclusion is that Biewers are their own separate breed. They differ from Yorkshire Terriers in that they are homozygous recessive for a piebald gene located on chromosome 20, and a large section of chromosome 5 that seems to be closely related to the chromosome 5 of the Havanese and Maltese breeds. How that section of chromosome 5 got there is a matter of great debate, because the lines giving rise to Biewers were supposedly purebred Yorkshire Terriers based on pedigree records. But then again, pedigree records from that time period (the 1980's and before) are based on a breeder's honesty, since they preceded the era of DNA testing.
MARS Labs currently distinguishes between Yorkshire Terriers and Biewer Terriers in its purebred tests, and Biewer breeders are urged to breed their Biewers ONLY with other Biewers and not with Yorkshire Terriers.
I'm not as certain about DNA test results for Parti Yorkies, but I understand that at least some of them test as purebred Yorkshire Terriers, so they are distinct from Biewers, and probably arose from a spontaneous mutation of the coat color gene to piebald spotting within purebred Yorkshire Terrier lines.