You are correct. Traditional X middle-button behavior is to insert the primary selection, from any application. It is distinct from the cilpboard(s), has no "copy" step, and doesn't behave the same as "paste" (it can't paste over another primary selection).

Many people don't notice the difference, though, especially if they're used to an environment that lacks that functionality, like Windows. Some might even get confused or frustrated when the middle click doesn't work quite as they expect (which, of course, is because it's actually doing something different).

https://www.jwz.org/doc/x-cut-and-paste.html

It's also possible that OP remapped their middle button to paste from the clipboard, although their phrasing makes it seem unlikely.