A one-bin system in which you put trash and recyclables in a single container is sent to a mixed waste processing facility (aka. a “dirty MRF”). Mixed waste processing facilities comprise less than 5% of all material recovery facilities (MRFs) in the US. Estimates for how effective mixed waste processing facilities are at pulling out recyclables range from 5-30%.
Consider talking with your hauler about how you could support increased diversion and how the hauler would characterize the waste management activity. A conservative approach would be to track it as disposal and focus behavior change efforts on reducing overall waste generation.