Save to Lightroom from Photoshop as action

Unfortunately, Photoshop actions can't directly save files back to Lightroom. Instead, follow these steps:

1. Import images into Lightroom.
2. Right-click and "Edit in Adobe Photoshop."
3. Edit in Photoshop.
4. Use "Save" or "Save As" in Photoshop.
5. Changes will automatically appear in Lightroom.

No need for an action.

Just so we're clear: Photoshop doesn't save "back" to Lightroom. It just saves to disk. The files aren't "in" Lightroom. This is a common misunderstanding.

The reason it appears in the Lightroom catalog is that a background script tracks the file and adds it to the catalog.

For this to work, Lightroom has to remain open during the save process. 

I can't see how an additional script can make any difference one way or the other here. This is just a perfectly normal "Save" from Photoshop. At default settings, this will save back to the original folder the file came from. 

If Lightroom isn't open, it won't be visible in the catalog and you need to import it later.

@Chimli Retouch & Motion 

One option is a simple 1 step action, where you insert the menu item for Save.

Then File > Automate > Batch with the source being open files.

Otherwise, a custom script would be required – such as:

/*
Save all open docs.jsx
https://community.adobe.com/t5/photoshop-ecosystem-discussions/save-to-lightroom-from-photoshop-as-action/td-p/14201903
*/

#target photoshop

var currentDoc = activeDocument;
for (var i = 0; i < app.documents.length; i++) {
    activeDocument = app.documents[i];
    activeDocument.save();
}
activeDocument = currentDoc;

alert("Script completed!");

https://prepression.blogspot.com/2017/11/downloading-and-installing-adobe-scripts.html