The second LIGO–Virgo catalog of gravitational-wave (GW) transients has more than quadrupled the observational sample of binary black holes. We analyze this catalog using a suite of five state-of-the-art binary black hole population models covering a range of isolated and dynamical formation channels and infer branching fractions between channels as well as constraints on uncertain physical processes that impact the observational properties of mergers. Given our set of formation models, we find significant differences between the branching fractions of the underlying and detectable populations, and the diversity of detections suggests that multiple formation channels are at play. A mixture of channels is strongly preferred over any single channel dominating the detected population: an individual channel does not contribute to more than ≃70% of the observational sample of binary black holes. We calculate the preference between the natal spin assumptions and common-envelope efficiencies in our models, favoring natal spins of isolated black holes of ≲0.1 and marginally preferring common-envelope efficiencies of ≳2.0 while strongly disfavoring highly inefficient common envelopes. We show that it is essential to consider multiple channels when interpreting GW catalogs, as inference on branching fractions and physical prescriptions becomes biased when contributing formation scenarios are not considered or incorrect physical prescriptions are assumed. Although our quantitative results can be affected by uncertain assumptions in model predictions, our methodology is capable of including models with updated theoretical considerations and additional formation channels.