Currently available star cluster catalogues from the Hubble Space Telescope (HST) imaging of nearby galaxies heavily rely on visual inspection and classification of candidate clusters. The time-consuming nature of this process has limited the production of reliable catalogues and thus also post-observation analysis. To address this problem, deep transfer learning has recently been used to create neural network models that accurately classify star cluster morphologies at production scale for nearby spiral galaxies (D ≲ 20 Mpc). Here, we use HST ultraviolet (UV)–optical imaging of over 20 000 sources in 23 galaxies from the Physics at High Angular resolution in Nearby GalaxieS (PHANGS) survey to train and evaluate two new sets of models: (i) distance-dependent models, based on cluster candidates binned by galaxy distance (9–12, 14–18, and 18–24 Mpc), and (ii) distance-independent models, based on the combined sample of candidates from all galaxies. We find that the overall accuracy of both sets of models is comparable to previous automated star cluster classification studies (∼60–80 per cent) and shows improvement by a factor of 2 in classifying asymmetric and multipeaked clusters from PHANGS-HST. Somewhat surprisingly, while we observe a weak negative correlation between model accuracy and galactic distance, we find that training separate models for the three distance bins does not significantly improve classification accuracy. We also evaluate model accuracy as a function of cluster properties such as brightness, colour, and spectral energy distribution (SED)-fit age. Based on the success of these experiments, our models will provide classifications for the full set of PHANGS-HST candidate clusters (N ∼ 200 000) for public release.