However if I have this created through ADManager, it appears to make visible the whole path (like \\server01\profiles$\username).   

Is there any way to make it map the home drive (like to the H: drive letter) to their path at \\server01\profiles$\%username% but not show the full server name in the mapping?

I think the only way around this in the meantime is I have a Group Policy Preference that replaces the mapped H-Drive with custom path and label. Whereas I just use ADManager's home drive component for the folder creation and permission setting.