exporting

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.

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.