When Euan, a Brit, converses with Rebecca, an American, using sign language, he explains to the others that his uncle is deaf, which is how he learned to sign. But unless Euan's uncle is American, he would have been using -- and Euan would have learned -- British sign language, not American sign language. The two are not mutually intelligible, so Euan wouldn't really have been able to converse with Rebecca at all.
When Euan and Rebecca are introduced, he addresses Rebecca with her name sign and then introduces himself using his name sign. But that's not how 2 people would introduce themselves to each other using sign language. There's no universal sign for any given name, as the deaf community more or less "assigns" name signs to each other. In Euan's case, his deaf uncle would have most likely given him a name sign. At any rate, Euan wouldn't have known Rebecca's name sign; she would have had to show him first. Likewise, the correct way for Euan to introduce himself would have been to finger-spell his name and then add, "My name sign is _______."