Also do you have any advice as to how I restore the process of putting the .lrcat on Pictures?
Just use Time Machine to restore them in their old place. That's the advantage of Time Machine. If you find a
folder called 'Lightroom, 2018' (with inside it the 'Lightroom, 2018.lrcat' file and also something called 'Lightroom, 2018 Previews.lrdata', then restore that entire folder.
So do you agree with my statement concerning the catalog files or not?
Your basic understanding is correct, but you still struggle a bit with some definitions. I didn't want to go into detail because it was irrelevant for the question how to restore the catalog, but now that you ask me to, here are my comments:
Let me try this a different way. I do have backups of all the pictures on separate, additional remote drives. I can look at the original import of any picture as well as its edited version as I copy them at both stages, But those pictures are only part of the story. The actual LR catalog also contains the information about the changes I have made in editing. This information is what is supposed to be in Pictures. Since I have chosen to keep my actual pictures in remote drives to save space on my hard drive and create redundant files, they have been removed from the Pictures file but the information should still be under the catalog identification, e.g., "Lightroom, 2018.lrcat." In order to authorize a second computer for LR I must have a copy of that Lightroom, 2018.lrcat as well as the drive containing the pictures. So I have parts of a single LR catalog stored in two separate locations.
First of all 'Pictures' is
folder, not a
file. Secondly, you seem to think that the 'Lightroom catalog' is the combination of your pictures and your catalog file, hence your idea that you have "parts of a single LR catalog stored in two separate locations". Although this could be considered semantics only, that is not true. Your pictures are not considered to be a part of the Lightroom catalog. They are separate. The Lightroom catalog is just a single database file (that file with the .lrcat extension) and one or two support files with a .lrdata extension. They are normally stored in a folder carrying the same name, but that is not absolutely necessary. You could have multiple catalogs with their support files all in one and the same folder.
I often compare a Lightroom catalog with an address book. Your images are your friends. They live in their own houses (the folders where the images are stored). Your Lightroom catalog is your address book. When you say "all my friends are in my address book", then you obviously do not mean that they are physically inside that book somehow, but that you have all the information you need (addresses, phone numbers, birthdays) in that book. That's what the Lightroom catalog is too. It's an 'address book of your images", containing all the information (including edits) of your images, but not the images themselves.