I find this hard to believe. I do not think that American specifically count a certain way. I also believe that they are not taught to count in school, in fact, they are discouraged from counting with their fingers.
That said, I, an American, count three by holding up my thumb and and my index and middle finger like Germans. (I count 1 with index, 2 with index and middle, 3 with thumb, index, and middle, and 4 with index, middle, ring, and pinkie. All these numbers are with palm facing person I am talking to.)
I asked my mother how she counts three, she held up index, middle, and ring finger like the British guy in the movie. (She counts 1 with index, 2 with index and middle, 3 with index, middle, and ring, 4 with index, middle, ring, and pinkie. All these numbers were done with back of her hand facing the person she is talking too.)
My sister counts three with thumb, index, and middle finger. (She counts 1 with thumb, 2 with thumb and index, 3 with thumb, middle and index, four with index, middle, ring and pinkie. All was done with palm facing person she was talking too.)
That said, I find it hard to read into counting of fingers among Americans. However, British to German counting of three is possible. Assuming the characters assumed the British guy had a British accent, then you can read into that.
Or maybe it is referring to how people order in a bar. They may use different ways but that may be getting involved in bar culture.
That said, my family never goes to bars, so maybe we are an anomaly.