A precise measurement of the luminosity is a crucial input for many ATLAS physics analyses, and represents the leading uncertainty for cross-section measurements of W and Z bosons and of top quarks. The ATLAS luminosity measurement depends on the interplay of a set of complementary luminometers, based on specific subdetectors, such as LUCID-2, or the ATLAS tracking detectors and calorimeter subsystems. The LUCID-2 detector was used as reference luminometer in the second running period of the LHC (Run-2), and remains being used as such during the third running period (Run-3) which started in 2022. The luminosity calibration of LUCID-2, determined from van der Meer (vdM) scans during dedicated running periods in each year, requires a correction for standard physics data-taking conditions, as well as careful monitoring throughout the data-taking year. The track-counting algorithm, based on reconstructing and selecting tracks in the inner detector, is used for this purpose. The performance of the track-counting algorithm is studied in head-on collisions and beam separation scans in Run-3. Those scans are vdM-like beam separation scans performed under normal physics conditions distributed over the entire data-taking year. Studies calculating the correction of LUCID-2 with the track-counting algorithm from beam separation scans are performed and compared to the nominal correction extracted from head-on collisions. Additionally, the stability from track-counting with respect to the number of proton bunches in the LHC ring and the crossing angle of the proton beams at the interaction point within the ATLAS detector are investigated.