The law does not refer to vague "hate speech" (or a translation thereof) or extends any definition of illegal content. Instead it lists specific types of content for which there is decades of legal precedents. It extends the liability of platforms (which makes them more eager to delete borderline cases) but does not change anything about the legality of content itself.

(FWIW, I see many problems with that law and it should never have passed but there is so much misinformation being spread that it gets tiring to correct.)

No, now you are blatantly spreading misinformation yourself. The law has extended and modified the penal code.

https://de.wikipedia.org/wiki/Netzwerkdurchsetzungsgesetz

Just check 'verabschiedetes gesetz'.

You misread the Wikipedia article. That section compares a draft with the final law. Between those two versions the list of laws that social networks need to watch out for has been changed. The law has not modified the Criminal Code, nor was it ever planned. Please read carefully before accusing others of spreading misinformation or you will do it yourself.

Here are the full four pages of the law as passed: https://www.bmjv.de/SharedDocs/Gesetzgebungsverfahren/Dokume... and this is an English translation: https://www.bmjv.de/SharedDocs/Gesetzgebungsverfahren/Dokume...