What methods exist to calculate the ellipticity of galaxies
The first two methods mentioned in the question are really similar. They both use moments of the light distribution, only the weighting is different.

Another family of methods is 2D profile fitting to the image, with more or less sophisticated luminosity profile models. A simple case would be to fit a Sérsic profile. But instead, one could make a composite model with two or more elliptical profiles (say one exponential disk and a de Vaucouleur bulge). This would allow to separate the ellipticity of the disk from the ellipticity of the bulge. A widely used software for this is GALFIT by Chien Peng.