Depends on the plugin whether the presets are auto-populated or only show the ones you have used the plugin’s “hamburger” menu to get to the “Rename” menu item.
Always remember to advance to the next un-renamed item before making any new changes to the plugin - though you’ve renamed it, the changes will apply to that preset position. That is, renaming does not “lock in” the changes - it merely makes that preset position easier to remember what it does. It is the position that counts most to preserve the plugin settings.
So, yes, the list of Presets gets saved when you save the Rack. If using a “rack-less” plugin, the preset list gets saved with the song.
I hope I am being clear!