move catalog to a laptop with existing PSE 2020 prg installed.

Grace, I think you are still not understanding the difference between the catalog and your photo files.  Let me try to explain it this way:

• Think of your hard drive as an art museum containing your artwork.  The museum has different types of art such as Renaisance or Impressionist (folders).  There may be several galleries for each type of art such as Reubens or Picasso (sub-folders).
• Your catalog is like the museum's visitor's guide or catalog.  The guide does not contain the actual artwork.  It contains photo-representations of the art and/or names of the artists and their work.  Most importantly, it points you to where each gallery is to be found, and you can use it to browse through the museum's collections to find what you are interested in.
• If a painting is removed from the museum or moved to a different gallery, the museum's guide is useless to know how to find it unless the guide is updated.  And that is why it is essential to only use the Organizer to move, delete or rename a photo within the catalog.  If you use a different application like File Explorer to manage your photos, it is like they have been stolen and will be treated as missing by the Organizer (museum).
• If you use different catalogs on different hard drives to manage your photos, it is like using last year's museum guide to find the latest exhibit.

So, to answer your original questions:

1.   The most efficient, accurate and easiest way to work on two computers in two different locations is to place both your catalog folder and your photos on the same external hard drive which can be switched between the two computers.  Elements will then be using identical information on both computers.  Changes made to the catalog (such as adding or deleting photos, or tagging or editing them) will be made in one place (the shared EHD).
2.   You need to have the same version of Elements installed on both computers.  Catalogs are not compatible between different versions.  However, as I have said before, it does not matter which computer you use to restore the backup on the EHD.  Each computer will be able to access all of the necessary information on the hard drive.
3. I think it should now be clear that all of the photo/artwork MUST be on the shared external drive in order for the catalog to be in sync on both computers.

@Grace Muskokan said:  Why do I call the folder "My Photos" if it is not going to contain photos? 

That folder will contain photos (in sub-folders).  This is where the restore process will copy your original files to according to your existing hierarchy, if you follow my suggestion.

@Grace Muskokan said:  I thought at first the My Photos By Year, my collection of camera photos from 2005 onwards, had to be in this main top folder too.  . . .   Or do I put the Photo Backup by Year on the same EHD as the "My Photos" folder.

That will be done automatically by the restore process.  All of your original photos will be copied to the EHD in whatever location (folder) you choose.  If you choose one of your existing folder names as the name of the main folder, you will have the same name existing under it.  You can do this if you want to, but in my opinion this will be confusing.

@Grace Muskokan said: But now I think [the My Photos by Year folder] can stay on the other drive and I would just make it a Watched Folder. Is that right?

No!  Hopefully, I have already explained why this is incorrect.

@Grace Muskokan said:  As I understand it now I could have a small 2 TB EHD that only contains my catalog. It will have one folder on it called "My Photos" (though it could be called "My Catalog Location"). I use this EHD whenever I want to use PSE and plug it into my desktop or laptop. When I want to add photos I point it to the drive that has the stored Photo Backup by Year on it. The catalog will then create a second section on the catalog tree for photos from there.

You are suggesting that you use two EHDs - one for the catalog folder only and one for your original photos.  While this would work, there is no reason to set up this way.  The catalog folder takes up very little space.  It does not contain any of your photos.  (At most, it contains very small thumbnails of your photos.)  There is no reason to have it alone on a 2TB drive.  The folder for my catalog which contains more than a quarter million photos and videos takes up about 20GB. 

@Grace Muskokan said:  When I want to add photos I point it to the drive that has the stored Photo Backup by Year on it. The catalog will then create a second section on the catalog tree for photos from there.

If you want to add photos from a camera, card reader or scanner, the Organizer gives you tools to copy the files to any location you choose.  The catalog will then keep track of whatever hard drive and folder/sub-folder you choose.  It doesn't care where you store them.  It is only because you want to use the catalog on two computers, that you must place them on a hard drive that is shared by the two devices.

I hope my explanations have not confused you further.

I'm a little puzzled by your screenshot which shows two G drives.  I assume you are going to use the E drive as your shared drive.  As long as you know what is in each folder, it looks clean enough to use.  But are you using the Photo Backup By Year folder as your main folder to use for the restoration.  If so, you may end up with multiple folders with the same name.  I suggest something more basic like My Photos.

Thanks for your detailed responses and screenshots.  I think the best strategy is going to be to do a restore of the full backup onto one of the EHDs.  It really doesn't matter which one as long as space is not an issue.  Ideally, you would use perhaps a third EHD since you would then have a clean hierarchy structure without the mess that currently exists.  This will avoid confusion in the future and will be much easier to manage going forward.  Since the full backup exists on an EHD, it doesn't really matter whether you use your desktop or laptop to perform the Restore operation.

Here is what I would do in your situation:

• Create a new folder on the root of the EHD that is going to have the shared catalog and folders.  Name it something like My Photos. 
• In the following Restore Catalog dialog, choose the New Location option and Browse to the new folder you have created and select it.  Also select the Restore Original Folder Location option:

• After the Restoration has been completed, you should have all of your original photos located on the EHD.  The folder structure will still be a bit of a mess, but you can clean that up later using the click and drag method mentioned earlier.  Make sure that the My Folders panel is in Tree View.  You may have to right-click on the "My Photos"  folder and select Show All SubFolders from the context menu in order to see the entire hierarchy of folders.
• As I think you now already know, all management of your catalog photos must be conducted within the Organizer.   By this, I mean all file move, delete and rename operations.
• If all has gone as planned, your catalog folder which contains all database information for your files, including their location, tag and face recogntion information will also be under your newly created parent folder.  I recommend that you move this to a different location on the same EHD as the photos.  I'm assuming that the catalog folder is named My Catalog.  You can use Windows File Explorer to cut and paste this folder (and all of its contents including sub-folders) to a different location -  even the root of the EHD.   You will then be able to Browse to the My Catalog folder in the File>Manage Catalogs dialog on both computers.  You will need to select the Custom Location to do this.  Alternatively, once the restoration has taken place, on each computer simply double-click on the file named catalog.pse18db (found in the catalog folder) and the catalog should open in the Organizer.

One final point.  Make sure that whichever EHD you use, it has the same drive letter on both computers.

I still need some further clarification of what you have done, how the backups were made, where the original photos are etc.

The goal will be to put all of your original photos on an EHD which you can plug into your desktop and laptop computers with the same drive letter.  You will also need to put a single catalog folder on the EHD which you will then be able to open from either computer.  Changes made to the catalog will then be available on both computers.

So, here are my questions:

1.  Where were your original photos stored before you imported them into the catalog you are using on your desktop?  Are some or all of the original photos stored on your C drive?  (I notice that the jpg file in your first screenshot is in a deep folder hieararchy on your E drive and that it appears to be an original file.)
2. You say you have photos backed up on two EHDs.  What are the drive letters of the two EHDs?  Or do you only attach one at a time to your computer?
3. If the original files are on one of the EHDs, how do you back them up to the second EHD?  Do you simply copy them from one drive to the other, or do you use the Elements backup (or some third party backup software) to create that second backup?
4. You say you have a full backup that includes the necessary .buc and .tly files.  Do you believe this backup contains all of the files that you want in your Elements catalog?  And where is that backup stored?
5. The 2010 sub-folders are in two different Photo Backup by Year parent folders.  The parent folders are in different hierarchies and contain different months.  Can you explain how these folders were created.  Did you click and drag the folders in the My Folders panel of the Organizer or did you use some other method?  Either way, are all of the folders available in the catalog that you used to take the screenshots?
6. Ultimately, how do you want your photos to be organized on the hard drive?  Is it sufficient for the files to be stored under a simple hiearchy with a main parent folder such as My Pictures and sub-folders containing the date-taken names you have given them?  Or do you want the sub-folders organized under a hierarchy that also is divided into years?
7. You have two different catalogs - one on each computer.  How did you create the second catalog?  Did you restore a catalog backup from the first one?  Or did you import all of the photos from the EHD into the second catalog?

Depending on your answers, I may have some further questions.  In the meantime, here is an article that may be useful, since it is likely that you will be able to do a new Restore of a full backup.

I will be back in a few hours to give you full instructions what to do.  Don't worry - it's all fixable.

It is not sufficient for you to have just the catalog folder on your laptop.  You must also have all of the photos/videos within the catalog database stored on your external hard drive.  So, please confirm that your desktop catalog is pointing to the files stored on your EHD.  And I'm talking about the original files, not the B000 . . . files that are copied as part of the backup. 

And just to be clear:  The Organizer's catalog does not make copies of your files to some hidden location.  It merely points to the files wherever you have saved them on your computer's hard drive(s).  (If you import files from a camera, phone or memory card reader, using the Adobe Photo Downloader, the files will be copied to a location that you choose.)