I have to say up front that I've never measured these bikes, but my old Mu SL is about the same size at the Heilios XX (it replaced that bike), and it was definitely bigger than my Birdy in all dimensions. The cover fit tightly on the Dahon and loosely on the Birdy. The other thing is that the Birdy shrinks vertically as the wheels fold under the bike. Still, there isn't *that* much difference.
Does the Brompton really come in larger than the CarryMe or Traincle? That seems potentially like an error.
One consideration is that the Dahon is consistently easy to fold. The total folding time is about the same between the two bikes, but the Birdy sometimes likes to give you a hard time. On the other hand, it is less bangable.
In terms of a durable small folding performance bike that is a pleasure to ride, the Birdy is tops. But the Brompton is significantly smaller and the Helios XX is notably lighter out of the box. Japan offers such a great array of bikes, you can have a look when you get there.