Now I made some workaround. I created a separate bot user with bot and admin rights in order to avoid "You can't edit page" errors. I don't know where to set the edit permisson for unregistered bots.

With settings in localsettings.php:
$wgGroupPermissions['*']['edit'] = true;
$wgGroupPermissions['*']['read'] = true;
and the dedicated bot user replace.py began to work correctly. It asked me for the password at last.
Then I set it back to
$wgGroupPermissions['*']['edit'] = false;
$wgGroupPermissions['*']['read'] = false;
and the bot previously logged in continued to work.

The mistake was that I should have explicitly run login.py -family:local before beginning botwork.
For some reason the bot didn't ask for my password when the wiki was set to private, but it is possible that the reason was a config error in family file which I have corrected meanwhile; I don't know if these are connected or not.
So my problem seems to be solved now! :-)

Thanks for the help and sorry for putting unnecessary questions.

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Bináris