What is meant by the scholars' saying: "This was not mentioned in the Sunnah" or "It was mentioned in such-and-such," and does "mentioned" here refer to the accepted narration, which includes both the authentic (sahih) and the good (hasan)? And what is the difference between "mentioned" (warada) and "established" (thabata) in their saying?

The word "thabata" (ثبت - proven/established) indicates certainty in the authenticity of the transmission, whereas "warada" (ورد - it was reported/came) means it was transmitted or reached, and in the terminology of hadith (definition) scholars, it is considered one of the forms of weakening (sighat al-tamrid). As Imam An-Nawawi said: "If you wish to narrate a weak hadith without an isnad (definition) (chain of transmission), do not say: 'The Messenger of Allah (peace be upon him) said such and such...' Rather, say: 'It was narrated thus,' or 'It reached us thus,' or 'It was reported,' or 'It came,' or 'It was transmitted.'" The forms of weakening are used to indicate the weakness of a hadith or doubt in its authenticity, while the forms of certainty (sighat al-jazm) are used for authentic (sahih (definition)) or good (hasan (definition)) hadiths.