The ballistocardiogram (BCG) artifact is currently one of the most challenging in the EEG acquired concurrently with fMRI, with correction invariably yielding residual artifacts and/or deterioration of the physiological signals of interest. In this paper, we propose a family of methods whereby the EEG is decomposed using Independent Component Analysis (ICA) and a novel approach for the selection of BCG-related independent components (ICs) is used (PROJection onto Independent Components, PROJIC). Three ICA-based strategies for BCG artifact correction are then explored: 1) BCG-related ICs are removed from the back-reconstruction of the EEG (PROJIC); and 2-3) BCG-related ICs are corrected for the artifact occurrences using an Optimal Basis Set (OBS) or Average Artifact Subtraction (AAS) framework, before back-projecting all ICs onto EEG space (PROJIC-OBS and PROJIC-AAS, respectively). A novel evaluation pipeline is also proposed to assess the methods performance, which takes into account not only artifact but also physiological signal removal, allowing for a flexible weighting of the importance given to physiological signal preservation. This evaluation is used for the group-level parameter optimization of each algorithm on simultaneous EEG-fMRI data acquired using two different setups at 3T and 7T. Comparison with state-of-the-art BCG correction methods showed that PROJIC-OBS and PROJIC-AAS outperformed the others when priority was given to artifact removal or physiological signal preservation, respectively, while both PROJIC-AAS and AAS were in general the best choices for intermediate trade-offs. The impact of the BCG correction on the quality of event-related potentials (ERPs) of interest was assessed in terms of the relative reduction of the standard error (SE) across trials: 26/66%, 32/62% and 18/61% were achieved by, respectively, PROJIC, PROJIC-OBS and PROJIC-AAS, for data collected at 3T/7T. Although more significant improvements were achieved at 7T, the results were qualitatively comparable for both setups, which indicate the wide applicability of the proposed methodologies and recommendations.