When the Organizer is launched for the first time it will ask you whether you want to convert the last used catalog to the latest version. If your respond yes, it will first make a copy of your older version catalog and give it the old name with a numeric suffix, e.g. My Catalog will become My Catalog-1. I believe that catalog should remain in its original storage location and if you were to open the older Organizer, you will get a message that the catalog could not be found. That is because the catalog has been renamed. If you browse to the location of the -1 catalog, you should be able to open it.
The new converted catalog (which will have the original name) will be put in the default location which is the one accessible to all users.
You have not lost your second Elements 14 catalog. You first need to convert it to Elements 2023 format by going to File>Manage Catalogs and clicking on the Convert button. You will then be able to browse to its location where it should be found and you can convert it. Alternatively, look for the catalog.pse14db file in the second catalog folder and use a context menu to Open With Elements 2023. The catalog should then be automatically converted and opened in Elements 2023.
Added: I should have mentioned that you can move the catalogs to whichever storage location you desire. But it sounds like you already know how to do that. Also, the second catalog will go through the same conversion process as the first. A copy will be made and the folder will be given a numeric suffix.