First, you need to ensure that you're launching the plugin version of the software, not the standalone application.
Choose the Correct Option in Lightroom Classic
In Photo > Edit In, select the plugin version—not the option that appears at the top starting with "Edit in...".
Refer to the screenshot below for clarification:
After the editor opens, look at the top right corner:
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If you see Close and Save buttons, you're using the plugin version (correct).
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If you see Save and Save All, you've opened the standalone version, which is not intended for use as a Lightroom Classic plugin.
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What's next?
If you've confirmed that you're invoking the plugin correctly, the edited file has likely been re-imported, but it may not be visible if the Lightroom Classic Filmstrip is set to 'Previous Import'. This is a default setting that Lightroom Classic applies after importing images from a camera.
To view the edited file, you need to right-click on 'Previous Import' and select 'All Photographs' instead.
However, if you don't want to view your entire catalog, you can right-click on one of the source images and choose 'Go to Folder in Library.' This will take you directly to the folder where the original photo is located.
Alternatively, you can just click on the folder containing the original photo in the Catalog tab. This will re-focus Lightroom on that folder and display all the files that are currently in it. It's also possible to find the processed image in Catalog > All Photographs.