Is what the worshiper did, in performing the prostration of forgetfulness (sujood al-sahw) and considering the three rak'ahs as Witr, after forgetting a rak'ah in Tarawih prayer and then being reminded by his wife, correct? Or should he have completed the prayer and performed Witr as a single rak'ah?

Prostration of forgetfulness (sujud al-sahw) is valid for an accidental addition of a rak'ah, and the prayer becomes nullified by intentionally adding one. However, what you did does not suffice for Witr prayer, because it requires a specific intention when commencing it, and this did not occur. You were not obliged to act on your wife's statement regarding the addition of a rak'ah unless you became certain of it, in which case you would act upon your certainty.