It can mean a lot of other things in contexts. In fact, it conveys no more message than "Please say hello" as the literal meaning is "please represent me favourably to that person." People are very likely to use this in combination with phrases of gratitude or apology but the former does not express the latter by itself.
A request in Japanese to relate one's gratitude or apology to another person usually contains the exact wording that one wishes to say to them in person. Abstract nouns such as gratitude and apology are used in highly formal speeches such as diplomatic communiqués and sound phoney coming out from ordinary Joes like yours sincerely.
adexx, unless your audience is from the Ministry of Foreign Affairs, I suggest the following formats:
旨 is gist or the message that the listener/relater would takes from the speaker's utterance.
You can use -to to introduce the true utterance you would tell to the receiver of the message.