HI Susanne,
Which set of files you are using depends on your application and data quality. If the data you have is clean and the registration was accurate, there is little difference in the two set of meshes and it does not matter which one you take. Often, however, there is noise in the data, which is not explained by the fitted meshes after the registration. As you don't want to have the noise in the model, you should choose the fit and not the projection.
In cases where you really need to accurately represent the target surface, and the target surface is not too noisy, working with the projection is appropriate.
Best regards,
Marcel