Remember mirroring settings for specific displays

I figured it out, it was a Spaces/Exposé problem, not a monitor problem. Somehow my desktops got mixed up, and I had to reassign the black desktop to the AppleTV-Beamer to be able to black out the beamer image via cmd-F1 toggling from mirroring (while presenting) to extended monitor (while black). The monitor at home was not affected. Thanks for helping, Luis.

ps I didn't write what I want:

1. I'd like to turn the beamer black without using the remote. The cmd-F1 toggle with black second desktop does this fine. The menu bar doesn't disturb.
2. I'd like to be able to use the computer for something else than the presentation for a moment, without disconnecting from the beamer. With 1. I'd get this with a black screen, but ideally I'd like to choose what the audience should see while I do this.

thanks!

Interesting, it is good to know that it will do this automatically. I always struggle with a particular beamer that I access via wifi on an AppleTV. The beamer sometimes mirrors the macbook, as it makes sense for me, but it also sometimes connects as the main monitor (my external monitor at home is set up this way) which makes me search for the mouse forever.

Today the screen froze again as I toggled the presentation off and back on with cmd-brightness-down to show a second desktop with black wallpaper.

I can't figure out what is wrong or reproduce things well. At least it seems to be not an issue of the M1 not being able to distinguish various displays.

Do you have an idea what might happen here?

Thanks for your reply. I wasn't aware that cmd-F1 toggles mirroring vs. extended display. I thought it would toggle Desktop 1 and Desktop 2 in Spaces, so I gave Desktop 2 a black wallpaper and all was fine, until I got the new monitor. Since I had the new monitor, I still have Desktop 2 black but cmd-F1 would toggle to whatever I had on the other desktop at home.

The Apple TV is just there to reach the beamer on Wifi instead of an HDMI cable, to be more mobile. Do I maybe need to set up Spaces while connected to the Apple TV/ beamer? "Displays have separate Spaces" is on, maybe I need to turn this off. Help says it's required for Split View, but I use Moom or Window Tidy to arrange windows, not Split View, so I could do without. I'll try out next time I'm there whether turning separate spaces off brings my mirror/black toggle back.

In addition, it would be nice if I could leave the beamer on the presentation instead of black while using my computer for something else the audience shouldn't see. Is this possible without connecting as secondary monitor and moving cursors on a monitor behind my back?

no, I usually exit presentation mode for this and make the slides as big as possible. I use keynote.

Sometimes I need to eg. spontaneously search for a file and would like to black out the beamer while I search. This used to work fine outside of presentation mode with the black wallpaper on Desktop 2 and toggling with cmd-F1 between mirroring (for presentation) and black extended desktop (for going through files). If I just get this functionality back it would be very useful. Right now it just shows a desktop with my calendar and notes that I don't want to have on the beamer either.

This is a black screen I want, I do this deliberately, like turning the beamer off.

True, keynote would be built for presentation mode but I often switch to show things on the web or in a pdf, or take private notes during a break. So I'd rather use a cmd-F1-type solution where I can control what's projected.