Is celebrating the Prophet's birthday (Mawlid) with the intention of introducing the Prophet permissible or forbidden, if it does not coincide with the actual day of the Prophet's birth, and what is the ruling on using the word "Mawlid" for this event?

It is not prescribed to celebrate the birth of anyone, as it has not been mentioned in the Shari'ah; rather, it is taken from non-Muslims. The intent behind celebrating a "mawlid" (birthday) is to celebrate the day of a person's birth. As for speaking about the Prophet (peace be upon him), introducing him, and mentioning his noble qualities and Sunnah, this is desirable and prescribed at all times and is not called a "mawlid." There is no harm in people celebrating a marriage or circumcision, and for someone to preach to the people or speak about the Prophet (peace be upon him) during such occasions; this is prescribed and does not fall under the innovated celebration of a "mawlid." There is no harm in organizing an activity or gathering in the mosque to introduce the Prophet (peace be upon him) on any day, without singling out a specific day believing it to have special virtue. There is also no harm in offering food to those present, and it should be widely known that this is not called a "mawlid."