Unable to exit the recording session

Hi sjkmd

Am not sure which version of Keynote you may be using, or whether there is a Keynote update that may be more compatible with Monterey... generally a check for "Updates" in the App Store may reveal whether or not a Keynote update is waiting to be installed on your system.

Re: Is there another way to exit without losing the recording?

Seems it depends on whether or not you want to make use of the video in Keynote ....

eg: You can export the video to another format for safe-keepoing, if you wish.

Apple Support's online article from the current user guide gives various options when completing a recording: See: Record Keynote presentations on Mac - Apple Support

Eg: says:

To stop recording, click

To preview the recording, click .To play from a specific point, drag the playback slider at the bottom of your screen to where you want to start, then click .To edit the recording, do one of the following:

• • Re-record from a specific location: Drag the playback slider at the bottom of your screen to where you want to start recording, then click . Choose Record in the pop-up menu to begin recording.
  • Delete the entire recording: Click in the bottom-right corner of your screen, then select Clear.

Press Esc to exit the presenter display.

The recording is saved with your presentation and can be played in Keynote.

To export a recorded presentation as a video or another file format,

see Export to PowerPoint or another file format.

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Depending on the Keynote version you have installed (or update to):

the Current: Keynote User Guide for Mac - Apple Support may or may not be useful

eg:

There's a "Table of Contents" with links to various topics / articles.

All the best :-)