Since the update last week, I suddenly can't save my pics from PS back to LR classic, on an external hard drive. I don't want to be forced to lose my pics in the cloud, nor have to pay anymore additional fees.
Just to be clear: you're not saving to Lightroom. You're saving to disk, and a background script adds it to the Lightroom catalog so that it's accessible from Lightroom. If this background script doesn't run, you won't see the image in Lightroom, but it will still be on disk and accessible from, say, Bridge, or a file browser.
So first check in Bridge or a file browser that the file is correctly saved, to the correct location that you specify in Photoshop's save as dialog.
For the script to run, both applications need to be open during the entire save process. If you still don't see it in Lightroom, check the sort order. Maybe it's there, only put at the end of the filmstrip.
If still not there, there may be a problem with the script. Reinstalling Photoshop, then Lightroom, should repair it.
Check the following: Apple changed the security requirements in MacOS Monterey. Go to MacOS 'System Settings - Security & Privacy' and add Lightroom Classic and Photoshop to "Full Disk Access". Click on the lock icon to unlock the settings, then click on the plus icon to add Lightroom Classic and/or Photoshop if it isn't already listed. If it is listed, then just enable the checkbox. See also Quick Tips: How to give Full Disk Access to Lightroom Classic on macOS
This applies to Photoshop too, so that could the the reason you can't save to disk.