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Splash and lavender ARE separate genes, which is exactly why a bird can be both. That is like saying a radhead cannot have blue eyes, or a tall person cannot have a small jaw.
Let me phrase that differently. It sounds like you do not understand what a gene is and is not, and are mixing up different genes.
A gene is a protein string at a specific place (locus) on a chromosome. At that locus different ALLELES of the gene can occur. No other gene can occur at that locus--ONLY the alleles of that particular gene.
In the case of the blue gene, the alleles are blue (Bl) and not-blue (bl+). Bl dilutes whatever black pigment may be in the bird. bl+ does not dilute black pigment in the bird. Lavender is a different gene at a different locus. Its alleles are not-lavender (Lav+) and lavender (lav). Lav+ does not dilute pigment; lav dilutes all pigment: both black and red.
Notice that blue is a dominant gene (incomplete dominant we will save for another discussion) as indicated by the capital letter Bl, and that lavender is a recessive gene as indicated by the lower case letter lav. The + indicates wild-type--the allele found in the red jungle fowl. One can consider wildtype alleles as the default for each gene.
A dominant allele takes affect even if another allele is present; a recessive allele does not take affect if a more dominant allele is present. Only when there is no more dominant allele will it take affect.
The black gene is an entirely separate gene (and thus locus) from either lavender or blue, and really has no bearing on this particular discussion. People often say "black" when in actuality they mean "not-blue."