Is the wording "I will be among Your righteous servants, and I will try to fast on Mondays and Thursdays and pray the supererogatory (Sunnah) prayers" considered a vow (Nathr)? If so, is expiation (Kaffarah) required for not fulfilling it due to illness? And are there specific wordings for a vow?

The stated صيغة (formula) does not constitute a valid vow (نذر) because it is merely an informing and not a commitment. If a person makes a vow and then forgets what they vowed, they should strive to recall its type to the best of their judgment and fulfill it. If they are unable to identify its type, some scholars are of the opinion that all possibilities should be performed to definitively discharge the obligation. However, the majority of scholars hold that an expiation for breaking an oath (كفارة يمين) is required, because doubt about it is like not naming it, based on the Prophet, peace and blessings be upon him, saying: "The expiation for a vow if it is not named is the expiation for an oath."