The bug
If the Demo World is joined while the player is dead, the introduction popup appears twice: once before the "You Died!" screen, and another time after clicking "Respawn".
To reproduce
Run the game in Demo mode. (You can do this manually by specifying the correct arguments to the client.jar, or you can select a Mojang account that has not brought the game in the Launcher.)
Click "Play Demo World" to generate/enter the demo world.
Once the world has loaded, enable cheats and run
/kill
On the death screen, click "Title Screen" (and then click "Title Screen" again)
In the main menu, click "Play Demo World" to reenter the Demo World
→ ✔ The Demo Mode introduction popup screen is shownClick "Continue Playing!"
→ ✔ The death screen is shownClick "Respawn"
→ ❌ After a second or two, the Demo Mode introduction popup screen is shown again
Notes
After respawning in Demo Mode, the introduction popup is shown: this is normal behaviour. The bug is that the popup is still shown when the death screen appears immediately after joining the world, so it seems to the player that the popup is shown twice in a row.
If you use
Esc
to close the first Demo Mode introduction popup, the death screen does not show at all. This may be a separate issue.
This issue has been fixed in 22w43a (presumably due to the screen/GUI-related changes). The introduction popup no longer displays before the death screen.