Accurate incipient fault (IF) identification in distribution networks (DNs) is crucial for improving the reliability of industrial electricity. Previous studies made great efforts on massive data-driven IF identification methods. However, there is difficulty in extensive IF data recording and collection in engineering practice, as IF is not easy to trigger the action of relay protection devices. Furthermore, attempts to collect IF data from multiple DNs in a centralized manner to increase sample size seem theoretically feasible, but can become impractical due to data privacy concerns among power utilities in different administrative regions. Therefore, a pretraining method is designed to learn general knowledge about various faults in DNs, providing small sample learning ability to address the challenge of small samples of IF. Then, a federated learning-based distributed training framework is devised to protect the data privacy of power utilities in different regions in collaboration model training. In addition, the imbalance of IF data distribution in different DNs in engineering practice is embedded in model learning by an adaptive weighting factor $\omega ^{I}$ to improve generalization performance. The above-mentioned contributions bring out the conception of the pretraining adaptive federated learning IF identification method (PTAFedIF). Case studies on DNs IF data show that PTAFedIF achieves 96.7$\%$ identification accuracy in small sample conditions, significantly superior to traditional IF identification methods. The embedded $\omega ^{I}$ in PTAFedIF improves the IF identification accuracy by 0.5$\%$.