At the very end, when he chooses to speak in French, he says "Madame, I'm David..." How would he say that in French? Whereas in English you'd usually say I am X, rather than I am called X, in French it's the other way around.
But in this context the 'I am' is so much more significant than just giving his name. It indicates what he is as much as who he is.
So, would he have said "Madame, je suis David"?