Is it meant that the intention in the heart is to utter it verbally and then say the opening Takbir (Takbirat al-Ihram)? And how should the intention be for the Tarawih, Witr, and Tawbah prayers?

The intention (niyyah) to do something means that a person brings to mind the will to perform that action. The intention for witr prayer is simply to intend to pray witr. If one does not do so, the obligation of witr does not fall away. As for Tarawih, out of caution, whoever wishes to pray it should intend it specifically. And for Salat al-Tawbah (prayer of repentance), it is necessary to specify the intention because it is a particular supererogatory prayer linked to a specific reason, which is committing a sin. Uttering the intention verbally is not prescribed; its place is the heart.