Determining the number of collisions delivered by the LHC inside the ATLAS detector is a key part of the ATLAS luminosity program. Several sub-detectors and algorithms are used, among which is the track counting method. This method assumes that the number of tracks reconstructed in the ATLAS Inner Detector is proportional to the number of simultaneous collisions, thereby providing a measurement of luminosity. Track counting is well understood in proton-proton collisions, but needs to be adjusted for Pb+Pb collisions, where the track multiplicity is dominated by rare head-on collisions. Potential improvements to the track counting method for heavy ions are investigated, in particular, a track-based event counting algorithm, where events are accepted if they have at least one track passing a set of track selection criteria. The performance of the new algorithm based on Pb+Pb collision data recorded by the ATLAS detector in 2018 and 2022 is compared to the reference luminometer LUCID-2. This work demonstrates the potential of the track-based event counting algorithm as a new luminometer for heavy ion collisions recorded by the ATLAS detector in Run 3 and beyond.