Mac saves files in more than one place

If I download an attachment from my email, it automatically is saved in Downloads folder and automatically opens up in Preview. Then I go to the top of the document in Preview, to select a different folder I want to move the document to. You also have the option of changing the name of the file here. My understanding was always that here you can rename the document (or not), without copying it, and then you move it to your desired location. Maybe I have it wrong.

I understand that when changing the name, you perhaps create a copy, so you can have a copy with the old name in Downloads and then a copy with the new name in the folder you selected to move it to. However, when I go through this process, sometimes even when I don't change the name of the file, I end up with a copy in the last folder I saved something to, as well as a copy in the folder I selected to move it to.

I can do a search on my Mac for the document and find it in three locations, the downloads, the folder I want it in, as well as another random folder - which I am beginning to think is often the last folder I saved something to. That doesn't seem right, unless I am fundamentally misunderstanding something.