Understanding what drivers look at is important for many applications, including driver training, monitoring, and assistance, as well as self-driving. Traditionally, factors affecting human visual attention have been divided into bottom-up (involuntary attraction to salient regions) and top-down (task- and context-driven). Although both play a role in drivers' gaze allocation, most of the existing modeling approaches apply techniques developed for bottom-up saliency and do not consider task and context influences explicitly. Likewise, common driving attention benchmarks lack relevant task and context annotations. Therefore, to enable analysis and modeling of these factors for drivers' gaze prediction, we propose the following: 1) address some shortcomings of the popular DR(eye)VE dataset and extend it with per-frame annotations for driving task and context; 2) benchmark a number of baseline and SOTA models for saliency and driver gaze prediction and analyze them w.r.t. the new annotations; and finally, 3) a novel model that modulates drivers' gaze prediction with explicit action and context information, and as a result significantly improves SOTA performance on DR(eye)VE overall (by 24\% KLD and 89\% NSS) and on a subset of action and safety-critical intersection scenarios (by 10--30\% KLD). Extended annotations, code for model and evaluation will be made publicly available.