How do I switch output in presenter-console?

In LibreOffice I have the problem that presenter-console shows the same on both monitors.
Is there a way to route the output deliberately?

Perhaps it is your second monitor settings. Try to make it as “extended” instead of “duplicate”.

On libreoffice then you can select “slide show → slide show settings → presentation displays”

That’s correct - you must not use “mirror display”. But once you set the two monitors to display independent screens (usually you can just press your laptop’s hotkey multiple times to switch between modes), you can configure which should be the “projector” for the audience and which one for the speaker using either the menu as in the answer above, or by pressing the corresponding “exchange” button in the speaker’s view. See e.g.