What is the ruling on someone who, out of forgetfulness, sat down in the first rak'ah of a supererogatory prayer without making the taslim, then stood up and performed the second rak'ah? And is "the one performing supererogatory acts is his own master" a general rule in supererogatory acts?

What you did is correct, and you must perform the prostration of forgetfulness (sujud (definition) al-sahw), because whoever forgets in their prayer is prescribed to perform the prostration of forgetfulness, whether the prayer is obligatory or supererogatory, due to the general saying of the Prophet, peace and blessings be upon him: "If one of you forgets, let him prostrate twice," and his saying: "If a man adds or subtracts (from his prayer), let him prostrate twice." As for the hadith (definition), "The supererogatory fasting person is his own master," the Shafi'is and Hanbalis use it as evidence that it is not obligatory to complete supererogatory acts of worship such as prayer and fasting after commencing them, unlike Hajj and Umrah (definition). In contrast, the Hanafis and Malikis believe it is obligatory to complete supererogatory acts if one has commenced them.