The birthday you missed by a day
The ninth was their birthday. You knew. You thought of it on the eighth and again at some point on the ninth, and the day went, and on the tenth you saw someone else's message in a group and realised. And now a message would have to say belated, which advertises that you forgot, and it's the eleventh, and by the fourteenth it's absurd, and so the birthday goes unmarked, which is worse than any of that.
This is the stretch at its purest, because the surcharge arrives in a single day. A birthday message on the day needs no explanation. A day late it needs "belated". Three days late it needs "belated" and "sorry" and a reason. And the belief that runs it is that late is worse than never, that a late message draws attention to the lapse while silence hides it.
Silence hides nothing. Your friend knows it was their birthday. They know who wrote and who didn't. A message on the twelfth that says "Belated happy birthday, I hope it was a good one, what did you do" tells them that you thought of them and that the thought reached the screen, which is the only thing a birthday message was ever for. The silence tells them the thought didn't, or that there was no thought, and they can't tell which, and most people assume the second.
Liu and colleagues' finding from chapter 6 applies here directly. A late message is a surprise, and surprise is where the appreciation comes from. The friend who gets three messages on the day and one on the twelfth remembers the one on the twelfth, not because it was late, but because it came when nothing was expected, and because it meant you'd thought of them on a day that wasn't the day. I'm reading their result into a situation they didn't test, but it fits.
The belated line, then, is a good line and not an embarrassing one. It's two lines like any other, and the word "belated" is the whole surcharge. No sorry, no reason, no "I'm terrible with dates". Just belated, and the wish, and a question. The question matters because it turns a lapse into a conversation, and a conversation is what a birthday message was trying to start.
And if the birthday was in March and it's September, the belated line is still a good line. "I've just realised I missed your birthday by half a year, which is a record even for me, how are you." That's not a confession. It's the pick-up, which chapter 21 is about, with a reason attached that both of you can laugh at. Late by a day and late by a year are the same move, and the move is to send something.