Claire, in all my anthropology courses (including ones specializing in kinship) and in all my reading of kinship texts for the anthropology classes I professionally tutor and TA at the collegiate level, I have to say I have never found your definition anywhere. I honestly don't know where you're getting it from, but it's not anthropology.
I don't want to beat this to death, but:
>>First cousin is either a male or female child of a sibling of your parent.
Yes.
>>A child of your mother's first cousin is still your first cousin but "removed" by that one generation.
No. My mother's first cousin is my first cousin once removed. That is because my mother's first cousin's grandparent is my great-grandparent -- that extra step is the "removal."
The child of my mother's first cousin is my second cousin. That is because my mother's first cousin's child and I share the same great-grandparent -- no removal.
Cousin relations and removals all come down to where you connect through a grandparent. That is the guiding principle to follow.
>>A cousin of your cousin is a 2nd cousin.
No. A cousin of my cousin is either my first cousin OR not a relative at all, depending on which parent of my cousin the relationship goes through.
This is complicated, but let's use fake names in case an example helps make it clear:
Say your mother, Florence, has two sisters, Doris and Lucille. The children of Doris and Lucille are your first cousins; I think you agree with that. So in this case, Doris and Lucille's children are first cousins, and they are /also/ your first cousins.
Here's where your "cousin of a cousin" idea fails. Let's say Doris and her husband, James, have two children, Joe and Andy. James' brother Max also has two children, Sally and Sarah.
Joe and Andy are your cousins because your mother and their mother are sisters. Joe, Andy, Sally, and Sarah are also each other's cousins -- through their fathers. But Sally and Sarah -- your cousins' cousins -- aren't /your/ relatives whatsoever because that relationship goes through James, not your blood aunt Doris. So your definition cannot be correct, because obviously not all of your cousin's cousins can be your cousins to any degree.
Does that make sense, or did I complicate it more?
>>A child of a cousin of your cousin is your 2nd cousin once removed.
No. You can see from the above example why this statement is incorrect.
I am not sure if we can get you to agree with us on this, but I can say that a) your definition is not the anthropological one and b) the system on Geni is not going to change to fit your definition. So you may just need to agree to disagree on this one. :)