We present an innovative and widely applicable approach for the detection and classification of stellar clusters, developed for the PHANGS-HST Treasury Program, an $NUV$-to-$I$ band imaging campaign of 38 spiral galaxies. Our pipeline first generates a unified master source list for stars and candidate clusters, to enable a self-consistent inventory of all star formation products. To distinguish cluster candidates from stars, we introduce the Multiple Concentration Index (MCI) parameter, and measure inner and outer MCIs to probe morphology in more detail than with a single, standard concentration index (CI). We improve upon cluster candidate selection, jointly basing our criteria on expectations for MCI derived from synthetic cluster populations and published cluster catalogues, yielding model and empirical selection regions (respectively). Selection purity (confirmed clusters versus candidates, assessed via human-based classification) is high (up to 70%) for moderately luminous sources in the empirical selection region, and somewhat lower overall (outside the region or fainter). The number of candidates rises steeply with decreasing luminosity, but pipeline-integrated Machine Learning (ML) classification prevents this from being problematic. We quantify the performance of our PHANGS-HST methods in comparison to LEGUS for a sample of four galaxies in common to both surveys, finding overall agreement with 50-75% of human verified star clusters appearing in both catalogues, but also subtle differences attributable to specific choices adopted by each project. The PHANGS-HST ML-classified Class 1 or 2 catalogues reach $sim1$ magnitude fainter, $sim2times$ lower stellar mass, and are $2{-}5times$ larger in number, than attained in the human classified samples.
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