About the sign-rank of the Minsky-Pappert function
There's the Minsky-Papert function, which is a depth-two formula, OR composed with AND, where the OR is of size $n^{1/3}$ and the AND is of size $n^{2/3}$. I.e., it's $OR_{n^{1/3}}\circ AND_{n^{2/3}}$. 

Then there's the particular communication complexity problem studied in the papers you linked, which is now a function of 2 $n$-bit strings $x$ and $y$, where $x$ is given to Alice and $y$ to Bob. This function is $OR_{n^{1/3}}\circ AND_{n^{2/3}} \circ (x \vee y)$, maybe more clearly expressed as


  $\bigvee _{i=1}^{n^{1/3}}\circ \bigwedge_{j=1}^{n^{2/3}} \circ (x_{ij} \vee y_{ij})$


This is the communication problem for which both papers you linked to prove a sign rank lower bound. If you look at Corollary 1.1, they're talking about this function, not the Minsky-Papert DNF.

Note that any communication problem can be represented as a matrix, by having $x$ index rows and $y$ index columns. The papers prove lower bounds on the sign rank of this matrix for the above communication problem. (Rank and sign rank being measures that make sense for matrices.) What Minsky and Papert showed was a sign degree (or threshold degree) lower bound on the Minsky-Papert DNF. Degree (and sign degree) are measures of Boolean functions.