Should I base my expiatory fasting during the White Days and the combination of rewards on the fatwa mentioned by you, or on the fatwa found on Sheikh Ibn Uthaymeen's website, which mentions the conditions for expiatory fasting and the impermissibility of combining it with the six days of Shawwal?

There is no contradiction between the fatwas of Sheikh Ibn 'Uthaymeen. He holds that fasting the six days of Shawwal is a restricted supererogatory act that requires a specific intention, and it is not permissible to combine its intention with making up for Ramadan fasts because it is subsequent to Ramadan. As for fasting other days, such as Ashura, the Day of Arafat, the White Days, and Mondays and Thursdays, the Sheikh permits combining the intention for them with making up for Ramadan fasts or expiations, because the purpose of the Law is achieved by combining them, and both rewards are obtained.