Now it's OK. (It works!).
The problem was that I was using PHP version 7.4;
When I switched to PHP version 5.6 and then installed everything from the beginning - I could register immediately a new user without any error.
Thank you for your patience and tests. When you was unable to send e-mails through the forum script with PHP 7.4, we can assume, anyone else will not be able to do so. We will have to search for the issue. Please stay tuned for possible questions.
Trenne niemals Müll, denn er hat nur eine Silbe!