Problem in actions

I feel that intuitively you should understand that when you specify another name during recording, the action is always going to use that specific filename. But if you just record saving to a specific folder w/o changing the filename, the action will use the name of the current doc as filename.

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Maybe that is not your only problem. I forgot what happens if you do multiple saves from the same document to the same folder. This may not create different filenames (much), but overwrite or ask to overwrite.

If that is a problem, you can save to different folders or look into using a script. If your two saves are for color and B&W (for example), it kinda makes sense to save to two folders anyway, or at least, it's not a huge hassle. If you want all files in one folder, use a multi-rename tool and add "B&W" to the B&W filenames later.

"so I cannot batch process"

You can actually use action for batch process but without save step which can be overwriten as shown below on screenshot, just check Override Action "Save As" Command in File > Automate > Batch.

For everything else just follow @Stephen Marsh instructions.

Please post a screenshot of the action with the save step expanded and the panel wide enough to view all recorded content.

Example:

Above you can see that one step recorded the change in name and the path to the desktop. The other step I didn't change the name of the recorded saved file but I did set the path to the desktop.

Sorry, I don't quite understand; in saving the file, I must go to "save as" to direct the photo to the folder of choice. Each time I play actions, it keeps rewriting that same file name. I don't quite understand how to set the save path. ( If I click on "save", it will rewrite the file in lightroom, that is not what I need.)

It's not a bug...

When recording the save step, don't change the filename, only set the save path.

In the Batch command, you need to check the box for "override action save commands" or words to that effect (sorry, I'm not in front of Photoshop).

https://helpx.adobe.com/au/photoshop/using/processing-batch-files.html#batch_process_files_in_nested_folders_into_different_formats