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In perception tasks of automated vehicles (AVs) data-driven have often outperformed conventional approaches. This motivated us to develop a data-driven methodology to compute occupancy grid maps (OGMs) from lidar measurements. Our approach extends previous work such that the estimated environment representation now contains an additional layer for cells occupied by dynamic objects. Earlier solutions could only distinguish between free and occupied cells. The information whether an obstacle could move plays an important role for planning the behavior of an AV. We present two approaches to generating training data. One approach extends our previous work on using synthetic training data so that OGMs with the three aforementioned cell states are generated. The other approach uses manual annotations from the nuScenes dataset to create training data. We compare the performance of both models in a quantitative analysis on unseen data from the real-world dataset. Next, we analyze the ability of both approaches to cope with a domain shift, i.e. when presented with lidar measurements from a different sensor on a different vehicle. We propose using information gained from evaluation on real-world data to further close the reality gap and create better synthetic data that can be used to train occupancy grid mapping models for arbitrary sensor configurations. Code is available at https://github.com/ika-rwth-aachen/DEviLOG.