The classification of variable objects provides insight into a wide variety of astrophysics ranging from stellar interiors to galactic nuclei. The Zwicky Transient Facility (ZTF) provides time-series observations that record the variability of more than a billion sources. The scale of these data necessitates automated approaches to make a thorough analysis. Building on previous work, this paper reports the results of the ZTF Source Classification Project (SCoPe), which trains neural network and XGBoost (XGB) machine-learning (ML) algorithms to perform dichotomous classification of variable ZTF sources using a manually constructed training set containing 170,632 light curves. We find that several classifiers achieve high precision and recall scores, suggesting the reliability of their predictions for 209,991,147 light curves across 77 ZTF fields. We also identify the most important features for XGB classification and compare the performance of the two ML algorithms, finding a pattern of higher precision among XGB classifiers. The resulting classification catalog is available to the public, and the software developed for SCoPe is open source and adaptable to future time-domain surveys.