Measurements of ejected electron energy distributions are used in conjunction with electron spin labeling techniques to probe the mechanisms by which He(2 $^{3}$S), He(2 $^{1}$S), and He(2 $^{3}$P) atoms are deexcited at Ar and Xe films adsorbed on a cooled Cu(100) substrate. The data for both surfaces contain features similar to those observed in gas-phase Penning ionization, indicating that ejection results, in part, from Auger deexcitation, i.e., surface Penning ionization. For Xe, however, additional features are observed that can be attributed to resonance ionization of an incident excited atom followed by neutralization of the resulting ${\mathrm{He}}^{+}$ ion through an interaction that involves neighboring Xe atoms in the film. Indeed, the Xe data provide an exceptional example of a surface at which Auger deexcitation and resonance ionization occur in parallel with one another, with a branching ratio that changes significantly as the internal energy of the incident atoms increases. The ejected electron yield from both Ar and Xe films is substantially higher than for clean Cu(100), indicating that such films might form the basis of an efficient thermal-energy helium metastable-atom detector.