We consider an adaptive network made of interconnected agents engaged in a binary decision task. It is assumed that the agents cannot deliver full-precision messages to their neighbors, but only binary messages. For this scenario, a modified version of the ATC diffusion rule for the agent state evolution is proposed with improved decision performance under adaptive learning scenarios. An approximate analytical characterization of the agents' state is derived, giving insight into the network behavior at steady-state and enabling numerical computation of the decision performance. Computer experiments show that the analytical characterization is accurate for a wide range of the parameters of interest.