Ooh...
Finally, after completely reinstalling Avast, the update date appeared.
First, the usual removal of Avast Antivirus via Windows Control Panel. Then restart.
Avastclear.exe (Uninstall Utility
https://www.avast.com/en-us/uninstall-utility) utility was used in Windows safe mode for complete deletion Avast Free Antivirus 20.3.2405 (thanks to bob3160).
Then restart again.
But even after all this, there were still a lot of Avast folders and files in the file system (in folders "C:\ProgramData"; "C:\Users\UserName\AppData\Local"; "C:\Users\UserName\AppData\Roaming"; etc). All Avast folders and files were also deleted manually, thanks to "Everything" app with brilliant fast (in milliseconds) search feature. If someone in a similar situation is going to delete Avast folders and files, do it very carefully, you should know what you are doing.
Then the offline installer was used for installation Avast Free Antivirus 20.3.2405. During new installation, the PC was disconnected from the Internet. (And the disconnect from the Internet was performed at the first step, before deleting Avast Antivirus via Windows Control Panel.)
Before the Internet connection was established, the "Invalid date" label appeared again. And I then thought that all this time to reinstall was wasted. But after the definitions update, everything was fine.
I hope that in the future will be no problems with the update date again. And I hope that the next version of the Avast Antivirus will fix this to ensure that this problem does not occur.
Thanks to everyone who answered and helped here.
Screenshot (Russian language, but you can see that the date is all right; before reinstalling was English, to make it more understandable to the people of this forum):