Help with creating a very simple action.

You can not do that with an action. Actions can not test if a file exists in the output path or generate file names. So it can create unique name and not overlay an existing file.  That will require a Photoshop Script.  Adobe supplies such a script that will use your action and save unique new file into the output path.  The Script does all the saves so you need to remove the save that is in your action.  Then  use menu File>Scripts>Image Processor...  and set the parameters for the script processing in its dialog.  If Image Process does not support saving the file type you want to save there is a more flexible Plug-in version you can download and install for free on the web. Image Processor Pro it supports more file types and has more save options then Image Processor script. It also a Photoshop Plug-in Script.  Once installed you use it from Photoshop's  Plug-in menu File>Automate>Image Processor Pro...

Its simply you  need a simpler action one without a save. One less step.

When recording don't change the filename, just set the path.

As suggested, if you expand the action to reveal the steps you'll likely see both filename and path recorded.

Please post the atn-file or a screenshot of the fully expanded Action in the Actions Panel.