I would like to draw the graphical model for the distribution in this question and capture as many independence assumption as possible. The distribution is:
Let $A = \{\text{flipping coin A}\}$ and $B\{\text{flipping coin B}\}$ and $C = \begin{cases} 1 & \text{both heads or both tails} \\ 0 & \text{otherwise}\end{cases}$. So this means that $A \perp\kern-5pt\perp B ,B \perp\kern-5pt\perp C, C \perp\kern-5pt\perp A $ but $A$ and $B$ are not conditionally independent when given $C$
So, would the following graphical model (belief network) be an accurate representation?
Yes this is correct. Notice that conditioning on $C$ provides some information about the result of $A$ and $B$. In fact, if $C=1$, then $A=B$, otherwise $A \neq B$, so of course $A$ and $B$ are not independent conditioned on $C$.
The information you're capturing in this graphical model is: $A$ and $B$ are independent and $C$ is constant when conditioning on $A$ and $B$.