You're both correct. Yes, it acts as a clipboard, for all practical purposes. But also it's a terminology thing.
If you'd open ICCCM, it uses name "CLIPBOARD" for a different selection atom: https://www.x.org/releases/X11R7.6/doc/xorg-docs/specs/ICCCM...
So in X11 the term "clipboard" (in a technical, not colloquial sense) is reserved for a different selection. And thus primary selection is not a clipboard.
If you'd open ICCCM, it uses name "CLIPBOARD" for a different selection atom: https://www.x.org/releases/X11R7.6/doc/xorg-docs/specs/ICCCM... So in X11 the term "clipboard" (in a technical, not colloquial sense) is reserved for a different selection. And thus primary selection is not a clipboard.