After a week of absence, I had to install first an update to Start11 and immediately after that an update (KB5022913) to Windows 11 on two of my Pcs.
After reboot, my system tray was invisible on both PCs, but by blindly clicking on the left side of the taskbar I was able to start the application from the tray. To check what was wrong I set Start11 Let Start11 enhance the taskbar to off and the taskbar started to behave as expected. So I thought that something went wrong during the Start11 update install. Reinstalling Start11 did not help. So I uninstalled the KB5022913. Taskbar started acting normally.
On Stardock forums, I saw that more people are having similar issues and you suggested a post where an early build of Start11 which tries to fix the issues is offered. So I reinstalled the MS KB5022913 and installed the Start11 1.4 297 build.
Even with that build, there are some issues but with the left-aligned medium-size taskbar (which I normally use) icon positions are normal and the tray is visible again. With small and large taskbars all bets are off (and it also depends if the taskbar auto-hide is off or on).
So I can live with my usual settings until the fix will be ready.