You’re right, there’s a failure to communicate, and as craftyjon suggested, it would be best to go through some basics first. There are also some FAQ articles about library management, especially Creating a new symbol library and a new symbol in KiCad 5 is relevant. It’s outdated but the basics haven’t changed. You should create a new library, store it inside the project and choose “Project” as the library table. A later KiCad version asks for the library table first, then the saving location. After that you will see your Project library with the name you gave it in the Add Symbol dialog library list. It’s empty, and you can create a new symbol there or copy existing symbols into it.
If you are still unsure – there doesn’t exist a Project library or libraries unless they have been created first; and they are in the same list with the Global libraries. You can’t tell the difference unless you can tell them apart by the library names or look into the library tables, which you can see in Preferences → Manage Symbol Libraries:
“Nickname” is what you actually see in the Add Symbol dialog list as the library name.