The barring allele is indicated with B; the non-barring allele with b+; when referring to a sex-linked gene, a - is used to indicate that lack of a locus for it in the female.
The quality of the barring does not depend on which parent it was inherited from but upon whether the bird carries one or two copies, and the presence or absence of the slow feathering gene. If present the barring will be sharper/crisper. If absent it will be fuzzier. This is not sex-linked and is completely independent of the barring gene.
A hen carries only one copy of all sex-linked gene; her sex chromosome is shorter than the sex chromosome of males.
Incompletely dominant genes such as barring have an intermediate appearance when present in one copy versus two or none. Thus a barred bird carrying two copies of the gene appears significantly lighter than a barred bird carrying one copy. Thus a cock with one copy of the gene has the same appearance in his barring as a hen. A cock carrying two copies appears lighter. A cock with one copy actually has one copy of barred and one copy of not-barred. About half his progeny will receive one of these and the other half the other. This is not determined by the offspring's gender.
On the other hand, a hen passes her W chromosome to her daughters; the genes carried on her Z chromosome are passed to her sons. (Not technically accurate, but it is an easy way to explain and understand.) If she is barred, all her sons will receive that gene from her. If she is not barred, they will receive that gene.
Black roo over barred hen is sex-linked. Black daughters and sons with one copy of barring.
Barred roo (two copies) over black hen is not sex-linked. All offspring will have one copy of barring.
Barred roo (one copy) over black hen is also not sex-linked. About half the offspring will be barred and about half will not.
Barred roo (two copies) over barred hen is sex-linked. Males carry two copies; females one, so males appear lighter.
Barred roo (one copy) over barred hen is not sex-linked--exactly. All males will receive at least one copy (from mama); about half of the chicks, regardless of gender, will receive a copy of barring from papa; about half will not. So, some males will have two copies, some males will have one copy, some females will have one copy and some will be black.