I am writing about Rabbi Margalit’s answer regarding a convert’s birthday I would like to know: What kind of "celebration" of the birthday are you addressing/recommending?

There are no rules. It is proper to rejoice with family or friends and say Divrei Torah on the occasion. One can wear something new and say the Shehecheyanu Bracha. It is mainly a day for self-examining what one did during the past year and making resolutions. Rabbi Ro'i Margalit