We use an optical cavity in the regime of intermediate coupling between atom and cavity mode to detect single moving atoms. Degenerate polarization modes allow excitation of the atoms in one mode and collection of spontaneous emission in the other while keeping separate the two sources of light; we obtain a higher confidence and efficiency of detection by adding cavity-enhanced Faraday rotation. Both methods greatly benefit from coincidence detection of photons, attaining fidelities in excess of 99% in less than 1 mu s. Detailed studies of the second-order intensity autocorrelation function of light from the signal mode reveal evidence of antibunched photon emissions and the dynamics of single-atom transits.