How to fill cell-network partition matrix of a function?
You're pretty close.

I don't think that's what they intended as the partition matrix.  Instead, I think they intended the following matrix:

         x3
x1 x2 \  0 1
       -----
 0  0  | 0 0
 0  1  | 0 0
 1  0  | 0 0
 1  1  | 0 1


That's slightly different from what you have written.  In particular, the partition matrix will have $2^{k-1}$ rows and 2 columns.

It's best to think of the first sentence as one paragraph, and the remaining sentences as a second disconnected paragraph that is trying to explain a taxonomy for classifying a column of this matrix.  The author is proposing to classify every column into one of three types: all-zeros, all-ones, or contains both a 0 and a 1.  The paper represents an all-zeros column by 0, an all-ones column by 1, and the third case by X.  Thus, X is used to represent any column that contains at least one 0 and at least 1, i.e., any column that is not all-zeros and not all-ones.

By this classification, the partition matrix above would be of the form 0X: the first column is all-zeros (type 0), and the second column is type X.