Which hydrophobicity scales are best for detecting transmembrane regions, and why?
This has been studied by some of my labmates in Why is the biological hydrophobicity scale more accurate than earlier experimental hydrophobicity scales? I am not involved in their research, but here is the gist of the paper:

Different scales are, as you say, developed based on different criteria. In particular, Eisenberg scale is one of the consensus scales, that is hopefully at least as good as all of its components. It was believed that there were no big performance differences between scales. This could be why Uniprot doesn't specify their choice of scale: at the time, it didn't matter too much. A study in 2009 showed that the new UHS scale was actually better than others.

The results of the benchmark are that the differences are indeed not so big: roughly $\pm2\%$ in difference in prediction accuracy. The overall winner? Hessa, scoring $1.29\ \sigma$ over the average of the scales, but closely followed by UHS, PM1D and PM3D. Eisenberg performance, by the way, is average.

Final note: they found that a near optimal window size for all the scales was 21 with double the weight in the central third.