saving multi edits to LR

I am also still unsure what you mean by-

"sending from LR to PS for further editing then back to LR..sometimes want to keep more than one edit."

Is that- more than one version of the Photoshop Tif/PSD? Or More versions in Lightroom-Classic (ie. Virtual Copies)?

For saving extra copies with different edits of a TIF/PSD in Photoshop- you would select the TIF/PSD and "Edit a Copy" as per screen-clip then apply your re-edits:

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.

Not sure what you mean by Back To LrC. Are you sending this image to Ps?

Create Virtual Copies of the original and send each to Ps and save to unique name.