From the photos the setup looks the same as on my Cub Cadet although that has a 10hp Kohler. My dynastart only has two terminals too, as far as I recall each has a single wire going to the regulator but one of them also has the live going to it from the ignition switch. So two terminals but three wires in total. From your photos it looks like the wires are connected including the live wire.
I’ve just had an issue with a tractor and it was the key switch that was no longer making a contact inside.