None of these will store the files in the Reaper Project Folder until I tell them to with the "Save As" commend when it will move or copy the content which has been moved into Reaper.
That's pretty standard practice for how DAW's work. Some of them (Cubase, Reaper) will offer to copy the files, some (Pro Tools) will automatically copy/move the files, and some will leave them alone. We're trying to support as many DAW's as possible (and has to work within the restrictions of Windows file permissions), therefore we have to take the safe approach with leaving the files in a common directory until another app removes them.