"When you Save in PS Elements the original image is not changed, just the preview of it that is shown in the origanizer"
Elements works like Photoshop, which means changes to the files in the Elements editor are indeed written to the files on your drive or in an edited copy (a version set), and the thumbnails are updated.
If you are working on raw files in the ACR plugin, yes, the raw file is not changed, but its sidecar xmp file is updated when you click on the 'Done' button. It's not so common in Elements because most of the time PSE users use the 'Open' button to further edit in the pixel editor and save a version set in jpeg, psd, tiff...
I am not a Mac user, but Mac experts will tell you that you can't make the organizer and Photos work together. If I remember well the trick is to save a file to your drive from the organizer or the editor, then let Photos 'import' the photo into its secret garden. Also, I have read that there are plugins in Photos to use PSE as an external editor. Please Mac experts, correct me if I am wrong.
My workflow seems to be working however I do not use Organiser. I first import the image to Mac Photos and then right click on the image in Photos to select the external editor (ie PSE 2020). I then edit it in PSE 2020 and save. It then changes the Photos image to that of the edited image (although it has an issue with using masks and a few other things:: I am currently just colourising old B&W photos so workflow fairly straight forward).
If you import into Photos from anywhere on the Mac it will save a new copy of that image in Photos I think by default therefore you will not see your edited version if it was opened from within the Adobesphere.
There is a setting in Mac Photos preferences to copy items to the Photos library if they are located elsewhere; if you untick this Photos should import the image as an alias and it may be that this setting will save your edited photo in both Photos and Organiser but I've not tried.