if the display persists after a Restart, and can be "arranged" in the arrangement pane, then some software or hardware somewhere is creating a profile for a display that is not actually present.
The Mac uses a system that reminds me of “Plug and play” to determine what "real" displays are connected, and what its capabilities are.
To get a Mac display to become active, you need the Mac to query the display, and the display to answer with its name and capabilities. Otherwise, the display will not be shown as present, and no data will be sent to the display.
There are additional ways you can set up a "fake" display profile, like DisplayLink as I mentioned. Then no actual query would be sent, because the information would already be available. This is the approach used for the built-in display -- no query is actually sent.
If you wanted to do that for some reason, apps like SwitchResX and similar would be used.
¿have you ever used such a program to manually adjust display capabilities?