Why not include private randomness in internal communication information definition?
I agree that the definition you suggest is more natural. However, this definition is equivalent to the definition without the private randomness, so I assume they omit the private randomness just to have a shorter notation.

To see the equivalence, observe that
\begin{eqnarray}
I(X; \Pi | Y, PR(Y) ) &amp;=&amp; I(X; \Pi, PR(Y) | Y ) - I(X; PR(Y) | Y ) \\
&amp; = &amp; I(X; \Pi, PR(Y) | Y ) \\
&amp; = &amp; I(X; \Pi | Y ) + I(X; PR(Y) | Y, \Pi ) \\
&amp; = &amp; I(X; \Pi | Y )
\end{eqnarray}
Here, the first and third equalities follow from the chain rule, the second equality follows since $X$ and $PR(Y)$ are independent even conditioned on $Y$, and the fourth equality follows $X$ and $PR(Y)$ are independent even conditioned on $Y$ and $\Pi$. 

The last independence takes a bit of effort to verify, but it basically follows from the rectangle property of protocols (and one also needs to note that conditioning on $Y$ severs the possible dependence of $X$ and $PR(Y)$ given $\Pi$). A bit more specifically, observe that conditioned $\Pi$ and $Y$, the set of values of $X$ and $PR(Y)$ that are consistent with $\Pi$ and $Y$ form a rectangle. Since $X$ and $PR(Y)$ started as independent variables, conditioning them on being in this rectangle preserves their independence.