Help with Photoshop action, script, or variable not sure

Could you use Batch Rename in Bridge?

When using variables and data sets you can not rename files. Also when using data sets you can export or save files only in PSD file format. There are other ways to append to file name but the question is whether you want word date, actual date or both as already asked by @Stephen Marsh 

You can also count on variables to add text and some other way to append whatever you want to file name while converting PSD to desired output format.

@Sheaffer 

If it was just renaming files, then there would be no need to open them into Photoshop. As you wish to add text to the image, you will indeed need to open the files and resave them in Photoshop.

Looking at your two sample images, this should be easy enough to script. Would it be acceptable to type the date details into a prompt once per batch, then those details would be applied to the entire batch of input JPEG images in that top-level directory/folder?

And just to be crystal clear, do you simply wish to only include " Date" at the end of the current filename, or the actual date string, such as " AUGUST, 2 2021" or whatever the date you enter is?

Please try the following script, saving as JPEG quality 12. As the example files appear to be from a template, I have also added the feature to remove photoshop:DocumentAncestors metadata from the processed files (which is not a bad thing in itself, however, it can bloat file sizes if there is excessive entries).

NOV 28 Downloading and Installing Adobe Scripts

/* 
https://community.adobe.com/t5/photoshop-ecosystem-discussions/help-with-photoshop-action-script-or-variable-not-sure/td-p/12439599
Help with Photoshop action, script, or variable not sure

Batch Date Stamp.jsx
v1.0
Stephen Marsh, 9th October 2021
*/

#target photoshop

if (app.documents.length === 0) {

    // The "Test if Cancel..." needs to be within a function
    (function () {

        var savedDisplayDialogs = app.displayDialogs;
        var inputFolder = Folder.selectDialog("Select the input folder containing JPEG files", "");
        // Test if Cancel button returns null, then do nothing
        if (inputFolder === null) {
            app.beep();
            return;
        }

        var inputFiles = inputFolder.getFiles(/\.(jpg|jpeg)$/i);

        // var outputFolder = inputFolder;
        var outputFolder = Folder.selectDialog("Select the output folder", inputFolder);
        // Test if Cancel button returns null, then do nothing
        if (outputFolder === null) {
            app.beep();
            return;
        }

        var dateString = prompt("Enter the date (Month, Day Year)", "MONTH, ## ####");
        // Test if Cancel button returns null, then do nothing
        if (dateString === null) {
            app.beep();
            return;
        }

        app.displayDialogs = DialogModes.NO;

        for (var i = 0; i < inputFiles.length; i++) {
            open(inputFiles[i]);
            var docName = app.activeDocument.name.replace(/\.[^\.]+$/, '');
            var saveFileJPG = new File(new File(outputFolder + '/' + docName + ' date' + '.jpg'));
            dateStamp();
            // As this appears to be created from a template
            deleteDocumentAncestorsMetadata();
            saveJPG(saveFileJPG);
            app.activeDocument.close(SaveOptions.DONOTSAVECHANGES);
        }

        app.displayDialogs = savedDisplayDialogs;

        app.beep();
        alert("Script Completed! Files saved to: " + "\r" + outputFolder.fsName);


        /************ Functions ************/

        function dateStamp() {
            var doc = app.activeDocument;
            // Ruler units
            var origRuler = app.preferences.rulerUnits;
            app.preferences.rulerUnits = Units.POINTS;

            // Doc resolution
            var origRes = app.activeDocument.resolution;
            app.activeDocument.resizeImage(null, null, 72, ResampleMethod.NONE);

            // Text layer, based on a 72ppi resolution
            var layerRef = doc.artLayers.add();
            layerRef.name = "Date Info";
            layerRef.kind = LayerKind.TEXT;
            var textRef = layerRef.textItem;
            var textColor = new SolidColor();
            textColor.rgb.red = 255;
            textColor.rgb.green = 255;
            textColor.rgb.blue = 255;
            textRef.color = textColor;
            textRef.contents = dateString;
            textRef.position = new Array(0, 0);
            textRef.font = 'Arial-Black';
            textRef.size = 50;
            textRef.useAutoLeading = true;
            textRef.justification = Justification.CENTER;

            // Ruler units for positioning on canvas
            app.preferences.rulerUnits = Units.PIXELS;
            // Align text to horizontal centre
            align2SelectAll('AdCH');
            // Align text to bottom
            align2SelectAll('AdTp');
            // move text layer down
            doc.activeLayer.translate(0, 42);

            // Restore original doc resolution
            app.activeDocument.resizeImage(null, null, origRes, ResampleMethod.NONE);
            // Restore original ruler units
            app.preferences.rulerUnits = origRuler;

            function align2SelectAll(method) {
                /*
                AdLf = Align Left
                AdRg = Align Right
                AdCH = Align Centre Horizontal
                AdTp = Align Top
                AdBt = Align Bottom
                AdCV = Align Centre Vertical
                */
                doc.selection.selectAll();
                var desc = new ActionDescriptor();
                var ref = new ActionReference();
                ref.putEnumerated(charIDToTypeID("Lyr "), charIDToTypeID("Ordn"), charIDToTypeID("Trgt"));
                desc.putReference(charIDToTypeID("null"), ref);
                desc.putEnumerated(charIDToTypeID("Usng"), charIDToTypeID("ADSt"), charIDToTypeID(method));
                try {
                    executeAction(charIDToTypeID("Algn"), desc, DialogModes.NO);
                } catch (e) { }
                doc.selection.deselect();
            }
        }

        function saveJPG(saveFileJPG) {
            var jpgOptions = new JPEGSaveOptions();
            jpgOptions.formatOptions = FormatOptions.STANDARDBASELINE;
            jpgOptions.embedColorProfile = true;
            jpgOptions.matte = MatteType.NONE;
            jpgOptions.quality = 12;
            app.activeDocument.saveAs(saveFileJPG, jpgOptions, true, Extension.LOWERCASE);
        }

        function deleteDocumentAncestorsMetadata() {
            // https://prepression.blogspot.com/2017/06/metadata-bloat-photoshopdocumentancestors.html
            // https://forums.adobe.com/message/8456985#8456985
            if (ExternalObject.AdobeXMPScript === undefined) ExternalObject.AdobeXMPScript = new ExternalObject("lib:AdobeXMPScript");
            var xmp = new XMPMeta(activeDocument.xmpMetadata.rawData);
            xmp.deleteProperty(XMPConst.NS_PHOTOSHOP, "DocumentAncestors");
            app.activeDocument.xmpMetadata.rawData = xmp.serialize();
        }

    })();

} else {
    alert('No documents should be open when running this script!');
}

A script can Process all jpeg files in a folder.  A script can rename files.  What date do want to add to add to the  Jpeg files image name? Today's Date. Your birth date? If its a constant date like today's date the files can just be renamed they do not even need to be opened. If you need to have duplicate file in the folder with the two file names the script can copy the jpeg file?  There is no need to encode a new jpeg generation and lose some image quality. You should learn about scripting photoshop.   You can use Photoshop Batch to add a date to an image file name also Image Processor Pro can add a Date however they would open the original  jpeg and saves as additional copy with the dated names. You would beed toe select all the jpeg files ins the bridge and uset the  bridge menu Tools>Photoshop>Batch... or Image Pocessor Pro...