What is the ruling on saying "Allah bil-khair" (May Allah grant goodness) and "Sabbahak/Massak Allah bil-khair" (May Allah grant you goodness in the morning/evening) in the Prophetic Sunnah, and which of them is more correct, and is either of them considered an innovation?

The Sunnah is to initiate with greetings before speaking. If someone says, "May Allah grant you a good morning," there is no harm in it. As for their saying "Allah bil Khair" (Allah with goodness), it should be avoided due to the ambiguity it contains and because it was not reported from the Salaf.