Are there proofs from the Qur’an and Sunnah that dispraise is not backbiting in the six categories mentioned in al-Aqeedah al-Tahawiyyah?

The two verses are not part of the متن العقيدة الطحاوية (Matn al-Aqida al-Tahawiyya), rather they are a versification of the six situations in which backbiting (Gheebah) is permissible. These are:

1. Seeking redress (التظلم): It is permissible for the wronged person to complain to someone who has authority or the ability to grant him justice. 2. Seeking help to change an evil (الاستعانة على تغيير المنكر): Informing someone who is expected to have the ability to remove an evil, with the intention of removing it. 3. Seeking a fatwa (definition) (الاستفتاء): It is permissible for a questioner to describe the situation of the person who wronged him to a Mufti (definition) in order to find a solution. However, it is safer to say, "What do you say about a man…" without specifying the person. 4. Warning and advising Muslims (تحذير المسلمين ونصيحتهم): Such as criticizing narrators and witnesses (jarh al-ruwat wa al-shuhood), consulting on matters of engagement or partnership, advising someone who frequents an innovator (mubtadi') or a wicked person (fasiq), and mentioning the state of a person appointed to a position who is not competent for it. 5. One who openly commits sin or innovation (المجاهر بفسقه أو بدعته): It is permissible to mention him only with what he openly declares. 6. Identification (التعريف): If a person is known by a nickname such as "al-A'mash" (the person with weak eyesight) or "al-A'raj" (the lame person), it is permissible to identify him by that without intending disparagement.