Split refers to having only one copy of a recessive gene. There is no obvious indication that the gene is present (because it is recessive)--you know it is because of your breeding.
A black split to lavender is genetically (Lav,lav). A regular black would be (Lav,Lav), and if the split had had a 2nd copy of lav (lav,lav) it would have been lavender rather than black.
Any creature can only pass the genes it possesses to its offspring.
A lavender bird (lav,lav) will always pass lav to its offspring.
A split can pass either of the two alleles (Lav & lav) that it carries; some will receive Lav and others lav. Statistically that should be about 50/50.
A non-split black will always pass Lav.
Does this help?