Years ago, I wrote a program that my family reunion use a computer to identify their relationship to other people at a reunion. One of the things I realized that made this program much simpler was that one's siblings were one's 0th cousins.

You make is sound so simple, especially if you say it quickly :-)
But seriously, I do like your suggestion:
When they talk about each other X would say "Y is my n th cousin r times removed".
Y would say "I am n th cousin r times removed to X".

Not as difficult as you are making it sound. If you have a common ancestor Z with children X & Y producing a generational column of ancestors then the children of X & Y are cousins. I imagine no-one has problems understanding this.
To illustrate it graphically, imagine a three column strip with the first column labelled as n, second column as X ancestral line & the third column as Y ancestral line.
Thus label Z as n=-1, X & Y as n=0, then the (first)cousins are n=+1. For each generation down n increments and you have the n th cousin of X & Y.
When you stop going down the X column but continue down the Y column r times then you have the n th cousin r times removed.
When they talk about each other X would say "Y is my n th cousin r times removed".
Y would say "I am n th cousin r times removed to X". thus no ambiguity.

Correction for Xth cousin: number of generations back minus one. For 2nd cousins, you go back 3 generations to find the common ancestor, a great-grandparent.

The Xth cousin part is how many generations you have to go back to find a common ancestor. The Yth removed part is how many "levels" you are off. Example: Chip is my 1st cousin once removed. His mother and I are first cousins, we have a common grandparent. Chip and I are "once removed" because it is two levels back to the common ancestor for me, but three levels back for him. Look up cousin on wikipedia -- they have pictures!

Hmmm -- it would seem that it must be commutative -- but that can make it a bit "lop-sided".
So if you are the first cousin to some one, and they have a kid and that kid has a kid, then the kid's kid is your first cousin twice removed. But similarly, you are that kid's kid's first cousin twice removed.
My head hurts...

So the answer didn't say explicitly, but I assume that the "x times removed" is commutative? In other words, I am the first cousin once removed of my mom's cousin....so she is my first cousin once removed also, right?

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