Does repeating a prayer due to doubt about the Taslim – while the first prayer was valid – invalidate the first prayer?

The first correct prayer is not invalidated by the second prayer. The first prayer thus becomes the obligatory one (fard), and the second becomes a supererogatory prayer (nafilah), based on the Prophet's (peace and blessings of Allah be upon him) saying: "If you have prayed in your dwellings, and then come to a mosque where a congregation is praying, then pray with them, for it will be a supererogatory prayer for you." Ibn Qudamah reported that this was the view of many of the early Muslims (Salaf) and the imams.