The website you linked to refers to piebaldism as "partial albinism". Piebaldism is an autosomal dominant genetic disorder caused by pathogenic variants of the KIT gene.
This is a fascinating disorder because the genetic change that causes the symptoms is present in all of the cells of the body, but the manifestations occur in relatively well demarcated areas.
Affected individuals demonstrate relatively stable areas of depigmentation of the hair and skin which are present at birth (Thomas et al 2004. PMID 15485525). Irregularly shaped white patches are frequently located on the face, trunk, or extremities and are usually symmetrical in distribution.
This image from the medical literature demonstrates one of the patches.
Figure 1 from Chiu et al 2012 available here.
Skin pigmentation is synthesized by melanocytes. At a high level, during embryogenesis, these cells are most likely generated in the neural crest located in the back of the embryo. These melanocyte precursors (melaonblasts) migrate forward and around the body.
The exact mechanism is not fully understood, but Oiso and colleagues note (2012. PMID 22670867):
The significant or complete loss of melanocytes in the ventral white patches and white hair is caused by improper migration of melanoblasts in embryo.
Presumably, the patches are symmetric because the loss of melanoblasts occurred in the neural crest and then failed to migrate around the embryo on either side. They seem to occur at the ventral midline because this is the farthest distance the melanoblasts have to migrate.
However, the patches are not always symmetric.
Figure 1 from Funkhouser et al 2019.
Unfortunately, our understanding of neural crest migration fails to explain how something like this could happen.
Funkhouser and colleagues note:
The long-standing dogma of melanocytic development and migration is that melanocytes are derived from neural crest cells, which originate in the neural tube and then migrate, proliferate, and differentiate into various tissues throughout the body after formation of the neural crest... While this pattern fits well with the patterns of certain congenital pigmentary abnormalities, particularly those following the lines of Blaschko, it does not explain other anomalies including large congenital melanocytic nevi nor the contiguous pigmentary patterning observed in our patients with piebaldism.
Thus, the short answer to your questions is, we aren't really sure.