What exactly happened in the restaurant in "All The Old Knives"?

In the closing scenes, happenings in the restaurant are intercut with scenes from the apartment, the CIA station, Celia's conversation with Vick (Lawrence Fishburne's character), and yet other times and places, which makes the dialogue perhaps difficult to pick out; but to answer your specific question, "what did Henry say which confirmed [it]", maybe this exchange will do:

Celia: Henry, Henry, listen to me.
C: They're gonna say that it was suicide.
C: You don't have much longer.
Henry: [realising how he has been poisoned] Oh, wine country.
C: I just don't understand why you did it.
C: Why, Henry?
H: I did it for you.
H: Because I love you.

We then cut to a flashback between Henry and

Ilyas

wherein the latter explains what happened to him since the last time Henry saw him, and how

he now has the power to harm Celia, who has gone to meet a contact, who (as it turns out) is in league with Ilyas; and therefore Henry is to do what he is told - to give up the CIA's secrets of Flight 127

Henry goes on to say, in the restaurant

H: I had to give him something. Ahmed's identity. I kept hoping they'd storm the plane, that they'd... release the prisoners. I made a choice, and I chose you.

Case closed.

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Also, why did the waitress mess up?

In the book, because she wasn't CIA, noticed the poison being administered, worked out what was happening, and became flustered. In the movie, I can't find specific subs related to this (beyond "We got the restaurant // Flying in a waiter and a bartender from DC"), although I have a (perhaps false) memory of such a scene

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_{(all subs from an SRT subtitle file I found on a janky website, although I don't feel too bad since I have paid for both the book and the movie)}