Lightroom Classic? or Lightroom? What version NUMBER?
In Lightroom Classic, I think you are either looking for a Virtual Copy, or a Snapshot. Either will allow multiple different edits of the same original photo. Virtual Copies will appear in the grid in the Library Module, so I prefer those.
Please note: there is no "save" in Lightroom Classic, you can't "save" a photo, and you should not be using the word "save" when in refers to a photo or your edits, and there is no "save" command. Your edits, virtual copies and snapshots are automatically stored in the Lightroom catalog without you taking any action.