The file, you need to copy and change the name of, is one of the first 17 files, i.e. one of those that end in .save, as the file ending in .save is the file the game will use to launch the save.
I’m guessing that the reason they are all named Slot_0000000
1 is because you’ve made backups on your hard disk or external hard drive and not because the game used to name the files differently (although I could of course be mistaken :) )? Each of these 17 Slot_00000001(various dates).saves has 5 backups that you see further down the list named from .ver.0 to .ver4 as SweetieWright82 mentioned. You don’t need to move these .ver files as well as they are just backups of the .save files. You can of course choose one of these to be the save you want to copy onto your new Sims 4 folder if you prefer that instead.
Anyway, before renaming your old file(s), you need to check what the files in your current save folder are called so that you don’t choose a “name” that is already being used. The game won’t be able read two files with the same name and may also overwrite one with the other. So if you already have a save called e.g. Slot_00000001.save, you can’t rename the file you want to transfer Slot_00000001.save, it has to be something different. I’ve read that some people say, the file name has to have 8 digits, but on this example from EA on how to restore a previous save, they actually go up to 9 in the example they provide (
https://help.ea.com/ca/help/the-sims/the-sims-4/how-to-restore-a-previous-saved-game-in-the-sims-4/). You can use digits (1-9) and in my folder I also have saves that end in a lower case letter. However, it’s stated here
https://answers.ea.com/t5/Technical-Issues-PC/How-do-I-combine-my-old-save-files-with-my-new-save-files/td-p/8911145 that the letters go from a to f - I’ve no idea how files are named beyond that.
Hope you manage to get your saves transferred so that they work :)
Oh, and you might very possibly need to copy your old tray file as well in order for the sims and builds from your old save(s) to show up. But someone else probably knows the answer to this better than me.