We present an innovative approach for stellar cluster identification and classification developed as part of the PHANGS-HST Treasury program, an NUV-U-B-V-I band imaging campaign for 38 nearby galaxies each having ALMA CO(2-1) maps. To distinguish cluster candidates from stars, we introduce the concept of a Multiple Concentration Index (MCI), defined as the average of several normalized CI values computed for adjacent radii pairs. Normalization is fixed to gauge how steep or flat sources appear compared to a fiducial cluster profile at the various radii being measured. Using two independent MCI measurements (one inner, one outer) allows us to probe source morphology in more detail than a single traditional CI. We improve upon cluster candidate selection, basing our criteria on MCI expectations derived from synthetic cluster populations tailored specifically to each galaxy and observation. Additionally, we define a universal selection polygon in MCI space which recovers the vast majority of clusters regardless of specific target/dataset. Selection purity (confirmed clusters versus candidates) is high (up to ~70% at typical cluster magnitudes for our sample) in the high-likelihood synthetic cluster MCI region / universal polygon, and somewhat lower overall (outside the preferred MCI region, or fainter). Use of machine learning (ML) classification accommodates the resulting large number of candidates in the later regime, allowing us to work significantly deeper than comparable surveys as result. We quantify the difference in cluster catalogs resulting from our approach versus prior surveys (e.g. LEGUS) and briefly explore trends in cluster age and mass with morphology.