Darcie,
In regards to the scientific binomial, you are correct, it is necessary to use both the Genus and species name.
Its an entirely different ball game with cultivars however, and listing just the cultivar name along with the Genus is perfectly acceptable, according to the ICBN regulations. It really doesn't matter what species it is, or is derived from, who first produced it, or the methods used. All that is essential is that the plant conform *exactly* to the registered description. Referring to your "white snake" example, if any "snake" out there matches your published description of "white snake" then it is that in fact. If it differs significantly, then it is not (and also has the potential to be registered in its own right. All that needs doing is to state the difference between it and the original "white snake")
The reason 'Albino' is correct vs. "alba" is because the ICBN rules do not permit Latinizations, which are reserved for the Latin binomials.
Also, if one wanted to be really cruel, they could technically register a blood red plant as 'Albino' and it would have to stand if so published.
There is a difference between horticulture and taxonomy. Taxonomy has always been the tool used to define and seggregate these plants, but this is not the only method possible, and I see the new use of registered cultivars as an additional tool that will help us communicate effectively about the plants we grow and distribute.