What is the correct form of seeking refuge (isti‘adhah) when reciting the Qur’an or commencing prayer, and is "A‘udhu billahi minash-shaytanir-rajeem" (I seek refuge in Allah from Satan, the accursed) impermissible?

What the Shaykh mentioned is incorrect. Seeking refuge with the words "Aūdhu billāhi minash-shayṭānir-rajīm" (I seek refuge with Allah from the accursed Satan) is a matter of consensus and is preferred by the majority of scholars. The matter of seeking refuge (istiādhah) is broad. The madhhab of the majority is that what is recommended is to say "Aūdhu billāhi minash-shayṭānir-rajīm" due to the apparent meaning of the verse: {So when you recite the Qur’an, seek refuge with Allah from Shaytan, the expelled [from His mercy]} (An-Nahl 16:98). Some scholars have recommended other additions, such as "Aūdhu billāhi As-Samīil-Alīm minash-shayṭānir-rajīm" (I seek refuge with Allah, the All-Hearing, the All-Knowing, from the accursed Satan). All of this is broad and good, but the best is "Aūdhu billāhi minash-shayṭānir-rajīm."