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If you always want JPGs from Photoshop-
Open the Preferences / External Editing, and set the "File Format" to JPG.
The procedure is then:
Select the image in LR
Right-click and choose "Edit In" -Photoshop.
When finished in Photoshop, Use "Save" (or simply Close the photo, and PS will ask you to 'Save")
*Do NOT use "Save AS".
Any time you use "Save AS" and change the file format ( eg. opened as TIFF, Save as JPG or opened as RAW and Save-AS a format different to LR preference) Lightroom does not know about this 'new' file you are creating.
As you note- any file will save back in the folder, and to see these 'new' files in LR you will need to simply- Right-click the FOLDER and "Synchronize Folder..." This will 'import' the 'new' image into the library.
Mohit's suggestion is another way to get the image into LR, by saving it in a folder that auto-imports, but you will need to set up that folder first as in his screen-clip, it is not available by default.