...
Error happens right after step, where I've entered database credentials.
This happens because initial /system/config.php file version with default values only is put into OpCache and therefore any content change of that file isn't reflect in sequential "require $file;" constructs for 2 seconds. Unlikely this is what happens during installation: change file content in one place > read it in another place.
Problem happens only when OPCache (see http://php.net/manual/en/book.opcache.php) is enabled in PHP.
Solution
When changing contents of "/system/config.php" file during installation in the "KInstallToolkit::SaveConfig" method, then the opcode cache for that file needs to invalidated.