Is saying "Allah by خير (Allah grant goodness)" after the salutation a correct phrase, or should it be avoided due to its ambiguity and because it was not transmitted from the Salaf, especially since there are previous fatwas indicating its correctness and others indicating that it should be avoided?

It is recommended to adhere to the Sunnah and initiate with greetings of peace. As stated in the Hadith: "Do not respond to one who speaks before greeting with peace." The greeting of the people of Islam is "Assalamu Alaikum," and it should not be replaced with other phrases, even if their meaning is correct. There is no contradiction between the fatwas of scholars regarding this matter. The phrase "Allah bil Khair" (May Allah grant you goodness) should be avoided because it was not transmitted from the Salaf (pious predecessors) and because it carries an ambiguity; it should not replace the Islamic greeting. The difference in wording between the fatwas is due to the varying formulations of the questions. The greeting of peace mentioned in the Sunnah is saying: "Assalamu Alaikum," and the response is: "Wa Alaikum Assalam," or "Salam Alaikum."