First of all, thank you very much for the various helpful feedback!
I tried out the inputs and was finally able to update the Joomla version to 4.4.1.

The language file and the databases were not the cause of the problem. I went through all the activated extensions again and deactivated some of them, e.g. Breezing Forms, a contact form file, Recaptcha and some extensions that were unknown to me but not system-relevant. I also set the "Joomla update" setting to "Joomla! Next" in the configuration. And oh wonderfully

the Joomla-Version update was available and could be implemented without problems. And finally, with this approch the PHP version 8.1. also works.
Thanks again, for the prompt support, you save my day. It is great in this forum when you realize that you're not left alone when you have problems.
Reto