While those definitions are true, “Sincerely” has become a type of hackneyed phrase that ends many letters to recipients. For example, it ends Cover Letters for jobs, Letters from banks, and other types letters. The word is similar to "Dear, _____" In my opinion, it has lost its "Deep-ness" throughout time.
If I were to define to add a meaning to it, it would be filler for the line before your name.
