The long-lost cousin's daughter would be a first cousin once removed and her kids would be a first cousin twice removed.
This is actually pretty timely as I've been looking up some long-not-seen cousin and have been contemplating the same question. Without the simple grandparent rule, I had to wonder what the order of operations would be.
Since my kids are first cousins once removed from my first cousin, whose kids are my first cousin's cousins once removed, are they first cousins twice removed [cousin(removed + removed)] or second cousins once removed [(cousin + cousin)removed] or second cousins twice removed (cousin + cousin)(removed + removed)
The grandparent rule makes it all a lot easier.

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?

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...

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!

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.

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.

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".

Great idea, and if one were a set of twins, the second could be x-squared. Triplets? x-cubed... "That is X**3, and over there is X**2, and that one is X. They are my power series!"

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.

One of my daughters married her fourth cousin. So, her children are not only her children, but her fourth cousins, once removed. My son-in-law's father is my third cousin, so these three (soon to be four) grandchildren through my daughter are also my third cousins twice removed through my son-in-law. I find that kind of fun!

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