An image is never "in" Lightroom. It's just at a certain location on disk, and your life will be much easier if you make a note of where that is. The lightroom catalog is just a database that keeps track of the file's physical location.
Once in Photoshop, a simple "Save" will save the file to whatever folder it came from. "Save As" gives you the option to choose destination folder.
If the file originally came from Lightroom, and Lightroom remains open during the whole process, a background script will again keep track of where the file goes, and add it to the catalog.
If it did not come from Lightroom, you have to import it into the Lr catalog. This doesn't move the file in any way, it just sits there. But the location will be recorded in the Lr catalog.
I'm back. If I hit Save after editing a photo in PS that did not come from LR, and then go to LR and click Import, how will LR know that the image I just Saved in PS is the one it should then import to LR?
Also...someone wrote that the way to do this is to Click 'Share' in PS and then choose LR and it will also appear in LR? True?
The easiest way to get that right is to import by dragging the new Photoshop file from the desktop, and dropping it in Lightroom Classic. The Import dialog box should come up with just that one file selected.
If you start out with an image in Lr, using Export will not make it return to Lr automatically. For that to happen, you have to use Edit in Photoshop. Then use Save in Ps, and the image will return to Lr. If you use Save As, the image will not return to Lr.