Is it permissible to complete a fast for a day when I intended to fast three consecutive days, believing the first day was Saturday, but then on Friday, after eating a morsel, I discovered it was actually the first day of the intended fast?

If food does not reach the throat, there is no harm in completing the fast. There is no requirement to form the intention for a supererogatory fast from the night before; rather, it is permissible to initiate it during the day. The proof for this is the hadith (definition) of Aisha, may Allah be pleased with her, and others, that the Prophet, peace and blessings be upon him, said: "Indeed, I am fasting then." And a number of the Companions also did so.