The key to understanding this is that there are two clipboards. The first clipboard is anything that is selected, e.g. selecting some text in a web browser. The contents of that clipboard are retrieved using the middle mouse button.
Choosing Cut or Copy from an application menu places whatever item was selected into the secondary clipboard, which can be retrieved using the corresponding Paste item on the destination application.
The two clipboards do not interact.
It's not exactly intuitive, but it is useful!"