Is there any way to force the Mac to not to log me out after ten minutes?

With this setting on the Mac Mini:

The screensaver comes on the screen turns off after ten minutes or so. This has just happened three times running.

Is there another setting which makes the Mac keep me logged in, with applications running?

Cleverly, it shuts down Apple Music, which loses its place (usually -- while reproducing the bug I left Safari foregrounded on the draft of this question, and the Mac was stopped short of killing Apple Music by the edit page's "discard" prompt -- but the screensaver still came on, with password). When I log back in, it's off the track I was listening to when the Mac lost the plot, and back on the default urgent advertisement for Taylor Swift.

Your power saver settings might be overriding this. Set them such that display is not turned off for a longer time.

https://support.apple.com/en-us/HT202824

Additionally, you should check "Logout after n minutes" in Security and Privacy-> General -> Advanced.